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Properties

Label 2-117-39.8-c1-0-3
Degree $2$
Conductor $117$
Sign $0.991 + 0.129i$
Analytic cond. $0.934249$
Root an. cond. $0.966565$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.404 − 0.404i)2-s + 1.67i·4-s + (1.88 − 1.88i)5-s + (−0.428 + 0.428i)7-s + (1.48 + 1.48i)8-s − 1.52i·10-s + (−1.88 − 1.88i)11-s + (3.24 + 1.57i)13-s + 0.346i·14-s − 2.14·16-s − 4.24·17-s + (−4.24 − 4.24i)19-s + (3.16 + 3.16i)20-s − 1.52·22-s − 5.39·23-s + ⋯
L(s)  = 1  + (0.285 − 0.285i)2-s + 0.836i·4-s + (0.845 − 0.845i)5-s + (−0.161 + 0.161i)7-s + (0.525 + 0.525i)8-s − 0.483i·10-s + (−0.569 − 0.569i)11-s + (0.899 + 0.435i)13-s + 0.0925i·14-s − 0.535·16-s − 1.02·17-s + (−0.973 − 0.973i)19-s + (0.706 + 0.706i)20-s − 0.325·22-s − 1.12·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 + 0.129i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.991 + 0.129i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.991 + 0.129i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28777 - 0.0836048i\)
\(L(\frac12)\) \(\approx\) \(1.28777 - 0.0836048i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 + (-3.24 - 1.57i)T \)
good2 \( 1 + (-0.404 + 0.404i)T - 2iT^{2} \)
5 \( 1 + (-1.88 + 1.88i)T - 5iT^{2} \)
7 \( 1 + (0.428 - 0.428i)T - 7iT^{2} \)
11 \( 1 + (1.88 + 1.88i)T + 11iT^{2} \)
17 \( 1 + 4.24T + 17T^{2} \)
19 \( 1 + (4.24 + 4.24i)T + 19iT^{2} \)
23 \( 1 + 5.39T + 23T^{2} \)
29 \( 1 - 6.40iT - 29T^{2} \)
31 \( 1 + (-3.10 - 3.10i)T + 31iT^{2} \)
37 \( 1 + (-1 + i)T - 37iT^{2} \)
41 \( 1 + (-6.13 + 6.13i)T - 41iT^{2} \)
43 \( 1 + 10.2iT - 43T^{2} \)
47 \( 1 + (1.88 + 1.88i)T + 47iT^{2} \)
53 \( 1 + 13.4iT - 53T^{2} \)
59 \( 1 + (-10.7 - 10.7i)T + 59iT^{2} \)
61 \( 1 - 6.48T + 61T^{2} \)
67 \( 1 + (-5.57 - 5.57i)T + 67iT^{2} \)
71 \( 1 + (5.32 - 5.32i)T - 71iT^{2} \)
73 \( 1 + (2.52 - 2.52i)T - 73iT^{2} \)
79 \( 1 - 3.05T + 79T^{2} \)
83 \( 1 + (2.23 - 2.23i)T - 83iT^{2} \)
89 \( 1 + (-3.50 - 3.50i)T + 89iT^{2} \)
97 \( 1 + (2.81 + 2.81i)T + 97iT^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.25578338775681149536130938811, −12.79463844732894515587312244380, −11.55374129519086561622756494092, −10.55805093612131586525354359017, −8.927602910564118474714157219259, −8.510467679408479782234938637795, −6.78176368214455181327391685210, −5.38762896790297927948027312313, −4.05379610776782390380675052542, −2.29198897628085916473172107633, 2.20268893634103564183822918399, 4.32781976995026468207965740066, 5.99844938935095761848973306701, 6.42634450596716633219427154184, 8.001634447643874090308002402921, 9.716605216990391631911641051160, 10.30438014495805918317087740451, 11.19400548057234795004684741095, 12.95453708608344162496108993242, 13.64553957500524752194787449435

Graph of the $Z$-function along the critical line