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Properties

Label 12-3800e6-1.1-c1e6-0-4
Degree 1212
Conductor 3.011×10213.011\times 10^{21}
Sign 11
Analytic cond. 7.80484×1087.80484\times 10^{8}
Root an. cond. 5.508465.50846
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·9-s − 6·11-s + 6·19-s + 2·29-s − 6·31-s − 18·41-s + 32·49-s + 20·59-s − 42·61-s + 2·71-s + 40·79-s + 24·81-s − 8·89-s − 48·99-s − 32·101-s − 14·109-s − 35·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 163-s + 167-s + 69·169-s + ⋯
L(s)  = 1  + 8/3·9-s − 1.80·11-s + 1.37·19-s + 0.371·29-s − 1.07·31-s − 2.81·41-s + 32/7·49-s + 2.60·59-s − 5.37·61-s + 0.237·71-s + 4.50·79-s + 8/3·81-s − 0.847·89-s − 4.82·99-s − 3.18·101-s − 1.34·109-s − 3.18·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 0.0783·163-s + 0.0773·167-s + 5.30·169-s + ⋯

Functional equation

Λ(s)=((218⋅512⋅196)s/2 ΓC(s)6 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(2-s)\end{aligned}
Λ(s)=((218⋅512⋅196)s/2 ΓC(s+1/2)6 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut &\left(2^{18} \cdot 5^{12} \cdot 19^{6}\right)^{s/2} \, \Gamma_{\C}(s+1/2)^{6} \, L(s)\cr=\mathstrut & \,\Lambda(1-s)\end{aligned}

Invariants

Degree: 1212
Conductor: 218⋅512⋅1962^{18} \cdot 5^{12} \cdot 19^{6}
Sign: 11
Analytic conductor: 7.80484×1087.80484\times 10^{8}
Root analytic conductor: 5.508465.50846
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (12, 218⋅512⋅196, ( :[1/2]6), 1)(12,\ 2^{18} \cdot 5^{12} \cdot 19^{6} ,\ ( \ : [1/2]^{6} ),\ 1 )

Particular Values

L(1)L(1) ≈\approx 5.2801115525.280111552
L(12)L(\frac12) ≈\approx 5.2801115525.280111552
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
5 1 1
19 (1−T)6 ( 1 - T )^{6}
good3 1−8T2+40T4−47pT6+40p2T8−8p4T10+p6T12 1 - 8 T^{2} + 40 T^{4} - 47 p T^{6} + 40 p^{2} T^{8} - 8 p^{4} T^{10} + p^{6} T^{12}
7 1−32T2+480T4−4261T6+480p2T8−32p4T10+p6T12 1 - 32 T^{2} + 480 T^{4} - 4261 T^{6} + 480 p^{2} T^{8} - 32 p^{4} T^{10} + p^{6} T^{12}
11 (1+3T+31T2+65T3+31pT4+3p2T5+p3T6)2 ( 1 + 3 T + 31 T^{2} + 65 T^{3} + 31 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
13 1−69T2+2089T4−35377T6+2089p2T8−69p4T10+p6T12 1 - 69 T^{2} + 2089 T^{4} - 35377 T^{6} + 2089 p^{2} T^{8} - 69 p^{4} T^{10} + p^{6} T^{12}
17 1−60T2+2020T4−41801T6+2020p2T8−60p4T10+p6T12 1 - 60 T^{2} + 2020 T^{4} - 41801 T^{6} + 2020 p^{2} T^{8} - 60 p^{4} T^{10} + p^{6} T^{12}
23 1−120T2+12p2T4−189357T6+12p4T8−120p4T10+p6T12 1 - 120 T^{2} + 12 p^{2} T^{4} - 189357 T^{6} + 12 p^{4} T^{8} - 120 p^{4} T^{10} + p^{6} T^{12}
29 (1−T+9T2−107T3+9pT4−p2T5+p3T6)2 ( 1 - T + 9 T^{2} - 107 T^{3} + 9 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}
31 (1+3T+57T2+105T3+57pT4+3p2T5+p3T6)2 ( 1 + 3 T + 57 T^{2} + 105 T^{3} + 57 p T^{4} + 3 p^{2} T^{5} + p^{3} T^{6} )^{2}
37 1−173T2+377pT4−657833T6+377p3T8−173p4T10+p6T12 1 - 173 T^{2} + 377 p T^{4} - 657833 T^{6} + 377 p^{3} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12}
41 (1+9T+103T2+563T3+103pT4+9p2T5+p3T6)2 ( 1 + 9 T + 103 T^{2} + 563 T^{3} + 103 p T^{4} + 9 p^{2} T^{5} + p^{3} T^{6} )^{2}
43 1−175T2+15366T4−820571T6+15366p2T8−175p4T10+p6T12 1 - 175 T^{2} + 15366 T^{4} - 820571 T^{6} + 15366 p^{2} T^{8} - 175 p^{4} T^{10} + p^{6} T^{12}
47 1−173T2+13777T4−738289T6+13777p2T8−173p4T10+p6T12 1 - 173 T^{2} + 13777 T^{4} - 738289 T^{6} + 13777 p^{2} T^{8} - 173 p^{4} T^{10} + p^{6} T^{12}
53 1−251T2+28142T4−1870607T6+28142p2T8−251p4T10+p6T12 1 - 251 T^{2} + 28142 T^{4} - 1870607 T^{6} + 28142 p^{2} T^{8} - 251 p^{4} T^{10} + p^{6} T^{12}
59 (1−10T+193T2−1172T3+193pT4−10p2T5+p3T6)2 ( 1 - 10 T + 193 T^{2} - 1172 T^{3} + 193 p T^{4} - 10 p^{2} T^{5} + p^{3} T^{6} )^{2}
61 (1+21T+265T2+37pT3+265pT4+21p2T5+p3T6)2 ( 1 + 21 T + 265 T^{2} + 37 p T^{3} + 265 p T^{4} + 21 p^{2} T^{5} + p^{3} T^{6} )^{2}
67 1−305T2+41985T4−3488601T6+41985p2T8−305p4T10+p6T12 1 - 305 T^{2} + 41985 T^{4} - 3488601 T^{6} + 41985 p^{2} T^{8} - 305 p^{4} T^{10} + p^{6} T^{12}
71 (1−T+132T2+139T3+132pT4−p2T5+p3T6)2 ( 1 - T + 132 T^{2} + 139 T^{3} + 132 p T^{4} - p^{2} T^{5} + p^{3} T^{6} )^{2}
73 1−65T2+9333T4−292257T6+9333p2T8−65p4T10+p6T12 1 - 65 T^{2} + 9333 T^{4} - 292257 T^{6} + 9333 p^{2} T^{8} - 65 p^{4} T^{10} + p^{6} T^{12}
79 (1−20T+292T2−2859T3+292pT4−20p2T5+p3T6)2 ( 1 - 20 T + 292 T^{2} - 2859 T^{3} + 292 p T^{4} - 20 p^{2} T^{5} + p^{3} T^{6} )^{2}
83 1−261T2+38793T4−3975273T6+38793p2T8−261p4T10+p6T12 1 - 261 T^{2} + 38793 T^{4} - 3975273 T^{6} + 38793 p^{2} T^{8} - 261 p^{4} T^{10} + p^{6} T^{12}
89 (1+4T+148T2+1067T3+148pT4+4p2T5+p3T6)2 ( 1 + 4 T + 148 T^{2} + 1067 T^{3} + 148 p T^{4} + 4 p^{2} T^{5} + p^{3} T^{6} )^{2}
97 1−429T2+86457T4−10512313T6+86457p2T8−429p4T10+p6T12 1 - 429 T^{2} + 86457 T^{4} - 10512313 T^{6} + 86457 p^{2} T^{8} - 429 p^{4} T^{10} + p^{6} T^{12}
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   L(s)=∏p ∏j=112(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{12} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−4.30421526036297072065023350168, −4.25921993980672892949589834630, −4.14169948379037910479312859877, −4.05181969328075046486303025632, −3.69211282472426697328912494887, −3.60662581388110246341105251411, −3.55297171408828834017597842906, −3.52026849561204917171305787522, −3.27285231820678467931274711395, −3.00900683175834972874577745819, −2.90061353124348997683568267497, −2.63909753318428756730844710575, −2.60205291545495832822974439348, −2.40862489649484407571771670361, −2.25296497467054431748179219903, −2.21941951284794359741144319074, −1.83616106002060038410262931476, −1.58310955005510152360629385083, −1.57440518974004092979092703762, −1.34930806915059113116715711415, −1.16512632568380846914502097557, −1.14088560416699761886137482040, −0.61399921636274559247474804549, −0.47451669956612729669134553388, −0.23519039623746656479886125353, 0.23519039623746656479886125353, 0.47451669956612729669134553388, 0.61399921636274559247474804549, 1.14088560416699761886137482040, 1.16512632568380846914502097557, 1.34930806915059113116715711415, 1.57440518974004092979092703762, 1.58310955005510152360629385083, 1.83616106002060038410262931476, 2.21941951284794359741144319074, 2.25296497467054431748179219903, 2.40862489649484407571771670361, 2.60205291545495832822974439348, 2.63909753318428756730844710575, 2.90061353124348997683568267497, 3.00900683175834972874577745819, 3.27285231820678467931274711395, 3.52026849561204917171305787522, 3.55297171408828834017597842906, 3.60662581388110246341105251411, 3.69211282472426697328912494887, 4.05181969328075046486303025632, 4.14169948379037910479312859877, 4.25921993980672892949589834630, 4.30421526036297072065023350168

Graph of the ZZ-function along the critical line

Plot not available for L-functions of degree greater than 10.