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Properties

Label 2-338-1.1-c7-0-77
Degree 22
Conductor 338338
Sign −1-1
Analytic cond. 105.586105.586
Root an. cond. 10.275510.2755
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 8·2-s + 12·3-s + 64·4-s + 210·5-s + 96·6-s − 1.01e3·7-s + 512·8-s − 2.04e3·9-s + 1.68e3·10-s − 1.09e3·11-s + 768·12-s − 8.12e3·14-s + 2.52e3·15-s + 4.09e3·16-s + 1.47e4·17-s − 1.63e4·18-s + 3.99e4·19-s + 1.34e4·20-s − 1.21e4·21-s − 8.73e3·22-s + 6.87e4·23-s + 6.14e3·24-s − 3.40e4·25-s − 5.07e4·27-s − 6.50e4·28-s − 1.02e5·29-s + 2.01e4·30-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.256·3-s + 1/2·4-s + 0.751·5-s + 0.181·6-s − 1.11·7-s + 0.353·8-s − 0.934·9-s + 0.531·10-s − 0.247·11-s + 0.128·12-s − 0.791·14-s + 0.192·15-s + 1/4·16-s + 0.725·17-s − 0.660·18-s + 1.33·19-s + 0.375·20-s − 0.287·21-s − 0.174·22-s + 1.17·23-s + 0.0907·24-s − 0.435·25-s − 0.496·27-s − 0.559·28-s − 0.780·29-s + 0.136·30-s + ⋯

Functional equation

Λ(s)=(338s/2 ΓC(s) L(s)=(− Λ(8−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}
Λ(s)=(338s/2 ΓC(s+7/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 338 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 338338    =    2⋅1322 \cdot 13^{2}
Sign: −1-1
Analytic conductor: 105.586105.586
Root analytic conductor: 10.275510.2755
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 338, ( :7/2), −1)(2,\ 338,\ (\ :7/2),\ -1)

Particular Values

L(4)L(4) == 00
L(12)L(\frac12) == 00
L(92)L(\frac{9}{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−p3T 1 - p^{3} T
13 1 1
good3 1−4pT+p7T2 1 - 4 p T + p^{7} T^{2}
5 1−42pT+p7T2 1 - 42 p T + p^{7} T^{2}
7 1+1016T+p7T2 1 + 1016 T + p^{7} T^{2}
11 1+1092T+p7T2 1 + 1092 T + p^{7} T^{2}
17 1−14706T+p7T2 1 - 14706 T + p^{7} T^{2}
19 1−39940T+p7T2 1 - 39940 T + p^{7} T^{2}
23 1−68712T+p7T2 1 - 68712 T + p^{7} T^{2}
29 1+102570T+p7T2 1 + 102570 T + p^{7} T^{2}
31 1+227552T+p7T2 1 + 227552 T + p^{7} T^{2}
37 1+160526T+p7T2 1 + 160526 T + p^{7} T^{2}
41 1+10842T+p7T2 1 + 10842 T + p^{7} T^{2}
43 1+630748T+p7T2 1 + 630748 T + p^{7} T^{2}
47 1+472656T+p7T2 1 + 472656 T + p^{7} T^{2}
53 1+1494018T+p7T2 1 + 1494018 T + p^{7} T^{2}
59 1+2640660T+p7T2 1 + 2640660 T + p^{7} T^{2}
61 1−827702T+p7T2 1 - 827702 T + p^{7} T^{2}
67 1−126004T+p7T2 1 - 126004 T + p^{7} T^{2}
71 1−1414728T+p7T2 1 - 1414728 T + p^{7} T^{2}
73 1+980282T+p7T2 1 + 980282 T + p^{7} T^{2}
79 1+3566800T+p7T2 1 + 3566800 T + p^{7} T^{2}
83 1+5672892T+p7T2 1 + 5672892 T + p^{7} T^{2}
89 1−11951190T+p7T2 1 - 11951190 T + p^{7} T^{2}
97 1+8682146T+p7T2 1 + 8682146 T + p^{7} T^{2}
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   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.746610323900973845929221721053, −9.214058044421978784169606051333, −7.83875535096920157374866104606, −6.80092325386742094856364294842, −5.77882889297338417001213472479, −5.21259867905917548975749469445, −3.43428918396002331413868064883, −2.97484543524493841640401803976, −1.62539192687212449331708170076, 0, 1.62539192687212449331708170076, 2.97484543524493841640401803976, 3.43428918396002331413868064883, 5.21259867905917548975749469445, 5.77882889297338417001213472479, 6.80092325386742094856364294842, 7.83875535096920157374866104606, 9.214058044421978784169606051333, 9.746610323900973845929221721053

Graph of the ZZ-function along the critical line