Dirichlet series
| L(s) = 1 | + 6·2-s + 15·4-s − 5-s + 10·7-s + 14·8-s − 6·10-s − 8·11-s + 8·13-s + 60·14-s − 21·16-s − 3·17-s − 7·19-s − 15·20-s − 48·22-s + 2·23-s + 9·25-s + 48·26-s + 150·28-s − 29-s − 14·31-s − 84·32-s − 18·34-s − 10·35-s − 7·37-s − 42·38-s − 14·40-s + 18·41-s + ⋯ |
| L(s) = 1 | + 4.24·2-s + 15/2·4-s − 0.447·5-s + 3.77·7-s + 4.94·8-s − 1.89·10-s − 2.41·11-s + 2.21·13-s + 16.0·14-s − 5.25·16-s − 0.727·17-s − 1.60·19-s − 3.35·20-s − 10.2·22-s + 0.417·23-s + 9/5·25-s + 9.41·26-s + 28.3·28-s − 0.185·29-s − 2.51·31-s − 14.8·32-s − 3.08·34-s − 1.69·35-s − 1.15·37-s − 6.81·38-s − 2.21·40-s + 2.81·41-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{12} \cdot 3^{36} \cdot 13^{12}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{12} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(24\) |
| Conductor: | \(2^{12} \cdot 3^{36} \cdot 13^{12}\) |
| Sign: | $1$ |
| Analytic conductor: | \(9.62433\times 10^{8}\) |
| Root analytic conductor: | \(2.36759\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(0\) |
| Selberg data: | \((24,\ 2^{12} \cdot 3^{36} \cdot 13^{12} ,\ ( \ : [1/2]^{12} ),\ 1 )\) |
Particular Values
| \(L(1)\) | \(\approx\) | \(23.70121980\) |
| \(L(\frac12)\) | \(\approx\) | \(23.70121980\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 2 | \( ( 1 - T + T^{2} )^{6} \) |
| 3 | \( 1 \) | |
| 13 | \( 1 - 8 T + 20 T^{2} + 27 T^{3} - 35 T^{4} - 1393 T^{5} + 8485 T^{6} - 1393 p T^{7} - 35 p^{2} T^{8} + 27 p^{3} T^{9} + 20 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} \) | |
| good | 5 | \( 1 + T - 8 T^{2} + 7 p T^{3} + 9 p T^{4} - 262 T^{5} + 697 T^{6} + 87 p T^{7} - 4054 T^{8} + 12043 T^{9} - 1077 p T^{10} - 37312 T^{11} + 161201 T^{12} - 37312 p T^{13} - 1077 p^{3} T^{14} + 12043 p^{3} T^{15} - 4054 p^{4} T^{16} + 87 p^{6} T^{17} + 697 p^{6} T^{18} - 262 p^{7} T^{19} + 9 p^{9} T^{20} + 7 p^{10} T^{21} - 8 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) |
| 7 | \( ( 1 - 5 T + 32 T^{2} - 138 T^{3} + 526 T^{4} - 1651 T^{5} + 4915 T^{6} - 1651 p T^{7} + 526 p^{2} T^{8} - 138 p^{3} T^{9} + 32 p^{4} T^{10} - 5 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 11 | \( 1 + 8 T - 8 T^{2} - 164 T^{3} + 276 T^{4} + 2812 T^{5} - 5870 T^{6} - 30636 T^{7} + 111620 T^{8} + 216104 T^{9} - 163164 p T^{10} - 1195244 T^{11} + 19752539 T^{12} - 1195244 p T^{13} - 163164 p^{3} T^{14} + 216104 p^{3} T^{15} + 111620 p^{4} T^{16} - 30636 p^{5} T^{17} - 5870 p^{6} T^{18} + 2812 p^{7} T^{19} + 276 p^{8} T^{20} - 164 p^{9} T^{21} - 8 p^{10} T^{22} + 8 p^{11} T^{23} + p^{12} T^{24} \) | |
| 17 | \( 1 + 3 T - 18 T^{2} + 183 T^{3} + 558 T^{4} - 3453 T^{5} + 21407 T^{6} + 29097 T^{7} - 375489 T^{8} + 2063736 T^{9} - 943650 T^{10} - 21149784 T^{11} + 153922011 T^{12} - 21149784 p T^{13} - 943650 p^{2} T^{14} + 2063736 p^{3} T^{15} - 375489 p^{4} T^{16} + 29097 p^{5} T^{17} + 21407 p^{6} T^{18} - 3453 p^{7} T^{19} + 558 p^{8} T^{20} + 183 p^{9} T^{21} - 18 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 19 | \( 1 + 7 T - 33 T^{2} - 72 T^{3} + 115 p T^{4} + 235 T^{5} - 39240 T^{6} + 207348 T^{7} + 654115 T^{8} - 3752154 T^{9} + 11663009 T^{10} + 60000606 T^{11} - 239857085 T^{12} + 60000606 p T^{13} + 11663009 p^{2} T^{14} - 3752154 p^{3} T^{15} + 654115 p^{4} T^{16} + 207348 p^{5} T^{17} - 39240 p^{6} T^{18} + 235 p^{7} T^{19} + 115 p^{9} T^{20} - 72 p^{9} T^{21} - 33 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 23 | \( ( 1 - T + 63 T^{2} + 25 T^{3} + 1816 T^{4} + 3686 T^{5} + 41135 T^{6} + 3686 p T^{7} + 1816 p^{2} T^{8} + 25 p^{3} T^{9} + 63 p^{4} T^{10} - p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 29 | \( 1 + T - 89 T^{2} - 8 p T^{3} + 3462 T^{4} + 13694 T^{5} - 79265 T^{6} - 358110 T^{7} + 1672508 T^{8} + 4477423 T^{9} - 46152024 T^{10} - 17793514 T^{11} + 1311898487 T^{12} - 17793514 p T^{13} - 46152024 p^{2} T^{14} + 4477423 p^{3} T^{15} + 1672508 p^{4} T^{16} - 358110 p^{5} T^{17} - 79265 p^{6} T^{18} + 13694 p^{7} T^{19} + 3462 p^{8} T^{20} - 8 p^{10} T^{21} - 89 p^{10} T^{22} + p^{11} T^{23} + p^{12} T^{24} \) | |
| 31 | \( 1 + 14 T + 2 T^{2} - 578 T^{3} + 657 T^{4} + 16815 T^{5} - 106001 T^{6} - 452680 T^{7} + 5755307 T^{8} + 11435717 T^{9} - 203168611 T^{10} - 275183188 T^{11} + 4884840401 T^{12} - 275183188 p T^{13} - 203168611 p^{2} T^{14} + 11435717 p^{3} T^{15} + 5755307 p^{4} T^{16} - 452680 p^{5} T^{17} - 106001 p^{6} T^{18} + 16815 p^{7} T^{19} + 657 p^{8} T^{20} - 578 p^{9} T^{21} + 2 p^{10} T^{22} + 14 p^{11} T^{23} + p^{12} T^{24} \) | |
| 37 | \( 1 + 7 T - 135 T^{2} - 1062 T^{3} + 10408 T^{4} + 85426 T^{5} - 548793 T^{6} - 4301988 T^{7} + 23576206 T^{8} + 139597497 T^{9} - 912959704 T^{10} - 2047091208 T^{11} + 34203967135 T^{12} - 2047091208 p T^{13} - 912959704 p^{2} T^{14} + 139597497 p^{3} T^{15} + 23576206 p^{4} T^{16} - 4301988 p^{5} T^{17} - 548793 p^{6} T^{18} + 85426 p^{7} T^{19} + 10408 p^{8} T^{20} - 1062 p^{9} T^{21} - 135 p^{10} T^{22} + 7 p^{11} T^{23} + p^{12} T^{24} \) | |
| 41 | \( ( 1 - 9 T + 216 T^{2} - 1503 T^{3} + 20385 T^{4} - 111492 T^{5} + 1080871 T^{6} - 111492 p T^{7} + 20385 p^{2} T^{8} - 1503 p^{3} T^{9} + 216 p^{4} T^{10} - 9 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 43 | \( ( 1 - 8 T + 110 T^{2} - 540 T^{3} + 7411 T^{4} - 38767 T^{5} + 407599 T^{6} - 38767 p T^{7} + 7411 p^{2} T^{8} - 540 p^{3} T^{9} + 110 p^{4} T^{10} - 8 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 47 | \( 1 - 13 T + 55 T^{2} - 992 T^{3} + 7731 T^{4} + 3223 T^{5} + 4148 p T^{6} - 715476 T^{7} - 16064455 T^{8} - 15214126 T^{9} - 274626885 T^{10} + 4257379366 T^{11} + 9843471887 T^{12} + 4257379366 p T^{13} - 274626885 p^{2} T^{14} - 15214126 p^{3} T^{15} - 16064455 p^{4} T^{16} - 715476 p^{5} T^{17} + 4148 p^{7} T^{18} + 3223 p^{7} T^{19} + 7731 p^{8} T^{20} - 992 p^{9} T^{21} + 55 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 53 | \( ( 1 - 6 T + 225 T^{2} - 1068 T^{3} + 23103 T^{4} - 92400 T^{5} + 1491289 T^{6} - 92400 p T^{7} + 23103 p^{2} T^{8} - 1068 p^{3} T^{9} + 225 p^{4} T^{10} - 6 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 59 | \( 1 + 4 T - 242 T^{2} - 964 T^{3} + 31107 T^{4} + 110189 T^{5} - 2882948 T^{6} - 7006719 T^{7} + 224872388 T^{8} + 256067587 T^{9} - 15934833126 T^{10} - 4535684977 T^{11} + 1006993338707 T^{12} - 4535684977 p T^{13} - 15934833126 p^{2} T^{14} + 256067587 p^{3} T^{15} + 224872388 p^{4} T^{16} - 7006719 p^{5} T^{17} - 2882948 p^{6} T^{18} + 110189 p^{7} T^{19} + 31107 p^{8} T^{20} - 964 p^{9} T^{21} - 242 p^{10} T^{22} + 4 p^{11} T^{23} + p^{12} T^{24} \) | |
| 61 | \( ( 1 + 28 T + 641 T^{2} + 9564 T^{3} + 123073 T^{4} + 1222640 T^{5} + 10688653 T^{6} + 1222640 p T^{7} + 123073 p^{2} T^{8} + 9564 p^{3} T^{9} + 641 p^{4} T^{10} + 28 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 67 | \( ( 1 + 4 T + 221 T^{2} + 42 T^{3} + 22060 T^{4} - 39769 T^{5} + 1645147 T^{6} - 39769 p T^{7} + 22060 p^{2} T^{8} + 42 p^{3} T^{9} + 221 p^{4} T^{10} + 4 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 71 | \( 1 - 13 T - 224 T^{2} + 2173 T^{3} + 44160 T^{4} - 240647 T^{5} - 5902487 T^{6} + 16066989 T^{7} + 606723725 T^{8} - 632364046 T^{9} - 51959470110 T^{10} + 254480426 p T^{11} + 3797519049485 T^{12} + 254480426 p^{2} T^{13} - 51959470110 p^{2} T^{14} - 632364046 p^{3} T^{15} + 606723725 p^{4} T^{16} + 16066989 p^{5} T^{17} - 5902487 p^{6} T^{18} - 240647 p^{7} T^{19} + 44160 p^{8} T^{20} + 2173 p^{9} T^{21} - 224 p^{10} T^{22} - 13 p^{11} T^{23} + p^{12} T^{24} \) | |
| 73 | \( ( 1 - 27 T + 546 T^{2} - 7342 T^{3} + 86442 T^{4} - 826815 T^{5} + 7577895 T^{6} - 826815 p T^{7} + 86442 p^{2} T^{8} - 7342 p^{3} T^{9} + 546 p^{4} T^{10} - 27 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
| 79 | \( 1 + 6 T - 321 T^{2} - 1214 T^{3} + 58899 T^{4} + 116916 T^{5} - 7829360 T^{6} - 6694632 T^{7} + 832965453 T^{8} + 193484266 T^{9} - 76900315647 T^{10} - 1630589046 T^{11} + 6387167408542 T^{12} - 1630589046 p T^{13} - 76900315647 p^{2} T^{14} + 193484266 p^{3} T^{15} + 832965453 p^{4} T^{16} - 6694632 p^{5} T^{17} - 7829360 p^{6} T^{18} + 116916 p^{7} T^{19} + 58899 p^{8} T^{20} - 1214 p^{9} T^{21} - 321 p^{10} T^{22} + 6 p^{11} T^{23} + p^{12} T^{24} \) | |
| 83 | \( 1 + 30 T + 171 T^{2} - 1950 T^{3} + 3123 T^{4} + 354252 T^{5} - 1712200 T^{6} - 34469640 T^{7} + 421745373 T^{8} + 3566743650 T^{9} - 39079405827 T^{10} - 46666748574 T^{11} + 4711712537310 T^{12} - 46666748574 p T^{13} - 39079405827 p^{2} T^{14} + 3566743650 p^{3} T^{15} + 421745373 p^{4} T^{16} - 34469640 p^{5} T^{17} - 1712200 p^{6} T^{18} + 354252 p^{7} T^{19} + 3123 p^{8} T^{20} - 1950 p^{9} T^{21} + 171 p^{10} T^{22} + 30 p^{11} T^{23} + p^{12} T^{24} \) | |
| 89 | \( 1 + 3 T - 219 T^{2} + 768 T^{3} + 21954 T^{4} - 162105 T^{5} - 28327 T^{6} - 1295874 T^{7} - 103218480 T^{8} + 2499980076 T^{9} - 8881578027 T^{10} - 139904365410 T^{11} + 2418191656953 T^{12} - 139904365410 p T^{13} - 8881578027 p^{2} T^{14} + 2499980076 p^{3} T^{15} - 103218480 p^{4} T^{16} - 1295874 p^{5} T^{17} - 28327 p^{6} T^{18} - 162105 p^{7} T^{19} + 21954 p^{8} T^{20} + 768 p^{9} T^{21} - 219 p^{10} T^{22} + 3 p^{11} T^{23} + p^{12} T^{24} \) | |
| 97 | \( ( 1 - 24 T + 687 T^{2} - 10501 T^{3} + 173697 T^{4} - 1929306 T^{5} + 22652301 T^{6} - 1929306 p T^{7} + 173697 p^{2} T^{8} - 10501 p^{3} T^{9} + 687 p^{4} T^{10} - 24 p^{5} T^{11} + p^{6} T^{12} )^{2} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{24} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−3.40453168405455254141270402560, −3.29632138206821163738020265076, −3.23545186781287349891374615723, −3.20855690952989045471990521727, −3.04459720143968739260164291937, −3.02501688476800966967828546453, −2.92818352800828954998604222229, −2.84621316664109082086221584093, −2.70448273811885117902272101263, −2.67546643137009814867394895985, −2.41170438519817638802656848106, −2.13480407492778309893260421520, −2.10411233835220365790235660269, −2.07269867619083156635079384348, −1.95125791440826028037979964548, −1.86987827244642355320166378113, −1.80537941766375432424002094947, −1.79093333326229060397518817209, −1.38139044814139201294142263564, −1.34214916166238180882237847222, −0.963082303481749466034264028663, −0.930156927767930818022403762178, −0.78825409799531290286345403704, −0.48577407988942257518676445052, −0.16484217043479372864982626791, 0.16484217043479372864982626791, 0.48577407988942257518676445052, 0.78825409799531290286345403704, 0.930156927767930818022403762178, 0.963082303481749466034264028663, 1.34214916166238180882237847222, 1.38139044814139201294142263564, 1.79093333326229060397518817209, 1.80537941766375432424002094947, 1.86987827244642355320166378113, 1.95125791440826028037979964548, 2.07269867619083156635079384348, 2.10411233835220365790235660269, 2.13480407492778309893260421520, 2.41170438519817638802656848106, 2.67546643137009814867394895985, 2.70448273811885117902272101263, 2.84621316664109082086221584093, 2.92818352800828954998604222229, 3.02501688476800966967828546453, 3.04459720143968739260164291937, 3.20855690952989045471990521727, 3.23545186781287349891374615723, 3.29632138206821163738020265076, 3.40453168405455254141270402560
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.