Dirichlet series
| L(s) = 1 | − 4·2-s − 3·3-s + 2·4-s − 8·5-s + 12·6-s − 7·7-s + 14·8-s − 7·9-s + 32·10-s − 18·11-s − 6·12-s − 13-s + 28·14-s + 24·15-s − 22·16-s − 2·17-s + 28·18-s − 6·19-s − 16·20-s + 21·21-s + 72·22-s − 22·23-s − 42·24-s + 17·25-s + 4·26-s + 34·27-s − 14·28-s + ⋯ |
| L(s) = 1 | − 2.82·2-s − 1.73·3-s + 4-s − 3.57·5-s + 4.89·6-s − 2.64·7-s + 4.94·8-s − 7/3·9-s + 10.1·10-s − 5.42·11-s − 1.73·12-s − 0.277·13-s + 7.48·14-s + 6.19·15-s − 5.5·16-s − 0.485·17-s + 6.59·18-s − 1.37·19-s − 3.57·20-s + 4.58·21-s + 15.3·22-s − 4.58·23-s − 8.57·24-s + 17/5·25-s + 0.784·26-s + 6.54·27-s − 2.64·28-s + ⋯ |
Functional equation
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(241^{7}\right)^{s/2} \, \Gamma_{\C}(s)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(2-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\left(241^{7}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{7} \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Invariants
| Degree: | \(14\) |
| Conductor: | \(241^{7}\) |
| Sign: | $-1$ |
| Analytic conductor: | \(97.7370\) |
| Root analytic conductor: | \(1.38722\) |
| Motivic weight: | \(1\) |
| Rational: | yes |
| Arithmetic: | yes |
| Character: | Trivial |
| Primitive: | no |
| Self-dual: | yes |
| Analytic rank: | \(7\) |
| Selberg data: | \((14,\ 241^{7} ,\ ( \ : [1/2]^{7} ),\ -1 )\) |
Particular Values
| \(L(1)\) | \(=\) | \(0\) |
| \(L(\frac12)\) | \(=\) | \(0\) |
| \(L(\frac{3}{2})\) | not available | |
| \(L(1)\) | not available |
Euler product
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ | |
|---|---|---|
| bad | 241 | \( ( 1 + T )^{7} \) |
| good | 2 | \( 1 + p^{2} T + 7 p T^{2} + 17 p T^{3} + 37 p T^{4} + 67 p T^{5} + 223 T^{6} + 327 T^{7} + 223 p T^{8} + 67 p^{3} T^{9} + 37 p^{4} T^{10} + 17 p^{5} T^{11} + 7 p^{6} T^{12} + p^{8} T^{13} + p^{7} T^{14} \) |
| 3 | \( 1 + p T + 16 T^{2} + 35 T^{3} + 110 T^{4} + 191 T^{5} + 467 T^{6} + 679 T^{7} + 467 p T^{8} + 191 p^{2} T^{9} + 110 p^{3} T^{10} + 35 p^{4} T^{11} + 16 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 5 | \( 1 + 8 T + 47 T^{2} + 38 p T^{3} + 132 p T^{4} + 1907 T^{5} + 5037 T^{6} + 11697 T^{7} + 5037 p T^{8} + 1907 p^{2} T^{9} + 132 p^{4} T^{10} + 38 p^{5} T^{11} + 47 p^{5} T^{12} + 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 7 | \( 1 + p T + 46 T^{2} + 4 p^{2} T^{3} + 786 T^{4} + 2528 T^{5} + 7897 T^{6} + 21047 T^{7} + 7897 p T^{8} + 2528 p^{2} T^{9} + 786 p^{3} T^{10} + 4 p^{6} T^{11} + 46 p^{5} T^{12} + p^{7} T^{13} + p^{7} T^{14} \) | |
| 11 | \( 1 + 18 T + 194 T^{2} + 1471 T^{3} + 8839 T^{4} + 43563 T^{5} + 182353 T^{6} + 651389 T^{7} + 182353 p T^{8} + 43563 p^{2} T^{9} + 8839 p^{3} T^{10} + 1471 p^{4} T^{11} + 194 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 13 | \( 1 + T + 43 T^{2} + 16 T^{3} + 74 p T^{4} + 171 T^{5} + 1275 p T^{6} + 3431 T^{7} + 1275 p^{2} T^{8} + 171 p^{2} T^{9} + 74 p^{4} T^{10} + 16 p^{4} T^{11} + 43 p^{5} T^{12} + p^{6} T^{13} + p^{7} T^{14} \) | |
| 17 | \( 1 + 2 T + 54 T^{2} + 118 T^{3} + 1511 T^{4} + 4204 T^{5} + 32789 T^{6} + 93345 T^{7} + 32789 p T^{8} + 4204 p^{2} T^{9} + 1511 p^{3} T^{10} + 118 p^{4} T^{11} + 54 p^{5} T^{12} + 2 p^{6} T^{13} + p^{7} T^{14} \) | |
| 19 | \( 1 + 6 T + 77 T^{2} + 408 T^{3} + 3328 T^{4} + 14851 T^{5} + 91875 T^{6} + 346087 T^{7} + 91875 p T^{8} + 14851 p^{2} T^{9} + 3328 p^{3} T^{10} + 408 p^{4} T^{11} + 77 p^{5} T^{12} + 6 p^{6} T^{13} + p^{7} T^{14} \) | |
| 23 | \( 1 + 22 T + 329 T^{2} + 3499 T^{3} + 30354 T^{4} + 215473 T^{5} + 1308858 T^{6} + 6746533 T^{7} + 1308858 p T^{8} + 215473 p^{2} T^{9} + 30354 p^{3} T^{10} + 3499 p^{4} T^{11} + 329 p^{5} T^{12} + 22 p^{6} T^{13} + p^{7} T^{14} \) | |
| 29 | \( 1 + 16 T + 228 T^{2} + 1945 T^{3} + 15113 T^{4} + 86708 T^{5} + 503802 T^{6} + 2527253 T^{7} + 503802 p T^{8} + 86708 p^{2} T^{9} + 15113 p^{3} T^{10} + 1945 p^{4} T^{11} + 228 p^{5} T^{12} + 16 p^{6} T^{13} + p^{7} T^{14} \) | |
| 31 | \( 1 + 18 T + 321 T^{2} + 3457 T^{3} + 35295 T^{4} + 269386 T^{5} + 1944797 T^{6} + 11129437 T^{7} + 1944797 p T^{8} + 269386 p^{2} T^{9} + 35295 p^{3} T^{10} + 3457 p^{4} T^{11} + 321 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 37 | \( 1 - 8 T + 140 T^{2} - 446 T^{3} + 5906 T^{4} + 5317 T^{5} + 140628 T^{6} + 725991 T^{7} + 140628 p T^{8} + 5317 p^{2} T^{9} + 5906 p^{3} T^{10} - 446 p^{4} T^{11} + 140 p^{5} T^{12} - 8 p^{6} T^{13} + p^{7} T^{14} \) | |
| 41 | \( 1 + 15 T + 165 T^{2} + 1716 T^{3} + 15349 T^{4} + 112837 T^{5} + 825907 T^{6} + 5652081 T^{7} + 825907 p T^{8} + 112837 p^{2} T^{9} + 15349 p^{3} T^{10} + 1716 p^{4} T^{11} + 165 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 43 | \( 1 - 14 T + 160 T^{2} - 1152 T^{3} + 8465 T^{4} - 39218 T^{5} + 236797 T^{6} - 1042279 T^{7} + 236797 p T^{8} - 39218 p^{2} T^{9} + 8465 p^{3} T^{10} - 1152 p^{4} T^{11} + 160 p^{5} T^{12} - 14 p^{6} T^{13} + p^{7} T^{14} \) | |
| 47 | \( 1 + 10 T + 213 T^{2} + 1823 T^{3} + 23033 T^{4} + 162514 T^{5} + 1573229 T^{6} + 9290969 T^{7} + 1573229 p T^{8} + 162514 p^{2} T^{9} + 23033 p^{3} T^{10} + 1823 p^{4} T^{11} + 213 p^{5} T^{12} + 10 p^{6} T^{13} + p^{7} T^{14} \) | |
| 53 | \( 1 - 15 T + 248 T^{2} - 2459 T^{3} + 26359 T^{4} - 226500 T^{5} + 2031262 T^{6} - 14891311 T^{7} + 2031262 p T^{8} - 226500 p^{2} T^{9} + 26359 p^{3} T^{10} - 2459 p^{4} T^{11} + 248 p^{5} T^{12} - 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 59 | \( 1 + 18 T + 283 T^{2} + 3342 T^{3} + 39787 T^{4} + 377400 T^{5} + 3440044 T^{6} + 26583077 T^{7} + 3440044 p T^{8} + 377400 p^{2} T^{9} + 39787 p^{3} T^{10} + 3342 p^{4} T^{11} + 283 p^{5} T^{12} + 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 61 | \( 1 - 4 T + 173 T^{2} + 229 T^{3} + 13815 T^{4} + 53354 T^{5} + 1189845 T^{6} + 3012271 T^{7} + 1189845 p T^{8} + 53354 p^{2} T^{9} + 13815 p^{3} T^{10} + 229 p^{4} T^{11} + 173 p^{5} T^{12} - 4 p^{6} T^{13} + p^{7} T^{14} \) | |
| 67 | \( 1 - 18 T + 312 T^{2} - 3657 T^{3} + 46177 T^{4} - 442394 T^{5} + 4388546 T^{6} - 34987571 T^{7} + 4388546 p T^{8} - 442394 p^{2} T^{9} + 46177 p^{3} T^{10} - 3657 p^{4} T^{11} + 312 p^{5} T^{12} - 18 p^{6} T^{13} + p^{7} T^{14} \) | |
| 71 | \( 1 + 50 T + 1452 T^{2} + 29886 T^{3} + 479876 T^{4} + 6256313 T^{5} + 67998420 T^{6} + 622620957 T^{7} + 67998420 p T^{8} + 6256313 p^{2} T^{9} + 479876 p^{3} T^{10} + 29886 p^{4} T^{11} + 1452 p^{5} T^{12} + 50 p^{6} T^{13} + p^{7} T^{14} \) | |
| 73 | \( 1 + 133 T^{2} + 1068 T^{3} + 10948 T^{4} + 119175 T^{5} + 1571649 T^{6} + 6028685 T^{7} + 1571649 p T^{8} + 119175 p^{2} T^{9} + 10948 p^{3} T^{10} + 1068 p^{4} T^{11} + 133 p^{5} T^{12} + p^{7} T^{14} \) | |
| 79 | \( 1 + 15 T + 468 T^{2} + 5553 T^{3} + 1243 p T^{4} + 946068 T^{5} + 12199714 T^{6} + 95010077 T^{7} + 12199714 p T^{8} + 946068 p^{2} T^{9} + 1243 p^{4} T^{10} + 5553 p^{4} T^{11} + 468 p^{5} T^{12} + 15 p^{6} T^{13} + p^{7} T^{14} \) | |
| 83 | \( 1 + 24 T + 705 T^{2} + 11938 T^{3} + 194891 T^{4} + 2473798 T^{5} + 28249166 T^{6} + 273618813 T^{7} + 28249166 p T^{8} + 2473798 p^{2} T^{9} + 194891 p^{3} T^{10} + 11938 p^{4} T^{11} + 705 p^{5} T^{12} + 24 p^{6} T^{13} + p^{7} T^{14} \) | |
| 89 | \( 1 + 13 T + 260 T^{2} + 2275 T^{3} + 16962 T^{4} + 2186 T^{5} - 807494 T^{6} - 17411725 T^{7} - 807494 p T^{8} + 2186 p^{2} T^{9} + 16962 p^{3} T^{10} + 2275 p^{4} T^{11} + 260 p^{5} T^{12} + 13 p^{6} T^{13} + p^{7} T^{14} \) | |
| 97 | \( 1 - T + 396 T^{2} + 357 T^{3} + 75395 T^{4} + 180194 T^{5} + 9562600 T^{6} + 26454385 T^{7} + 9562600 p T^{8} + 180194 p^{2} T^{9} + 75395 p^{3} T^{10} + 357 p^{4} T^{11} + 396 p^{5} T^{12} - p^{6} T^{13} + p^{7} T^{14} \) | |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{14} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−6.41021242033236520191815438528, −6.40155979517151252568185227180, −6.05697350071658060153813317514, −6.04538947049761488988295190963, −5.94676966412518300134568301165, −5.72573611436405750396965427389, −5.60925018361799856285197653623, −5.51461354208593135315591798741, −5.38892382368832931887551820150, −5.13631820784823267630632931308, −5.12427730936274790035134874150, −4.59652166428247929016248239880, −4.44359604513990188146003081844, −4.21671258110227393819275669874, −4.19330997205883169168635222870, −4.13876369370665627717071091564, −3.63381427959368206018850117794, −3.57888668523716984072157735305, −3.35815133163609533258214494515, −3.13621104260777994263964962567, −2.95217000379659306828558835759, −2.81935360930422956030879563752, −2.25453857323099446533056531089, −2.11090369589586914936954379992, −1.97523602169959753657160610366, 0, 0, 0, 0, 0, 0, 0, 1.97523602169959753657160610366, 2.11090369589586914936954379992, 2.25453857323099446533056531089, 2.81935360930422956030879563752, 2.95217000379659306828558835759, 3.13621104260777994263964962567, 3.35815133163609533258214494515, 3.57888668523716984072157735305, 3.63381427959368206018850117794, 4.13876369370665627717071091564, 4.19330997205883169168635222870, 4.21671258110227393819275669874, 4.44359604513990188146003081844, 4.59652166428247929016248239880, 5.12427730936274790035134874150, 5.13631820784823267630632931308, 5.38892382368832931887551820150, 5.51461354208593135315591798741, 5.60925018361799856285197653623, 5.72573611436405750396965427389, 5.94676966412518300134568301165, 6.04538947049761488988295190963, 6.05697350071658060153813317514, 6.40155979517151252568185227180, 6.41021242033236520191815438528
Graph of the $Z$-function along the critical line
Plot not available for L-functions of degree greater than 10.