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Properties

Label 2-117-117.11-c1-0-10
Degree $2$
Conductor $117$
Sign $0.0346 + 0.999i$
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  + (1.12 − 1.12i)2-s + (0.108 − 1.72i)3-s − 0.527i·4-s + (−1.83 − 0.492i)5-s + (−1.82 − 2.06i)6-s + (1.85 + 0.498i)7-s + (1.65 + 1.65i)8-s + (−2.97 − 0.376i)9-s + (−2.61 + 1.51i)10-s + (−1.22 − 1.22i)11-s + (−0.911 − 0.0573i)12-s + (2.63 + 2.46i)13-s + (2.64 − 1.52i)14-s + (−1.05 + 3.12i)15-s + 4.77·16-s + (−2.74 + 4.76i)17-s + ⋯
L(s)  = 1  + (0.794 − 0.794i)2-s + (0.0628 − 0.998i)3-s − 0.263i·4-s + (−0.821 − 0.220i)5-s + (−0.743 − 0.843i)6-s + (0.702 + 0.188i)7-s + (0.585 + 0.585i)8-s + (−0.992 − 0.125i)9-s + (−0.828 + 0.478i)10-s + (−0.369 − 0.369i)11-s + (−0.263 − 0.0165i)12-s + (0.731 + 0.682i)13-s + (0.708 − 0.408i)14-s + (−0.271 + 0.806i)15-s + 1.19·16-s + (−0.666 + 1.15i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0346 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0346 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.0346 + 0.999i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.0346 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04296 - 1.00743i\)
\(L(\frac12)\) \(\approx\) \(1.04296 - 1.00743i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.108 + 1.72i)T \)
13 \( 1 + (-2.63 - 2.46i)T \)
good2 \( 1 + (-1.12 + 1.12i)T - 2iT^{2} \)
5 \( 1 + (1.83 + 0.492i)T + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-1.85 - 0.498i)T + (6.06 + 3.5i)T^{2} \)
11 \( 1 + (1.22 + 1.22i)T + 11iT^{2} \)
17 \( 1 + (2.74 - 4.76i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.121 - 0.0326i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.99 + 5.18i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + 4.13iT - 29T^{2} \)
31 \( 1 + (1.98 - 7.42i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.76 + 1.00i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.0773 - 0.288i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.76 - 1.01i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (10.4 - 2.81i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + 9.44iT - 53T^{2} \)
59 \( 1 + (-5.60 - 5.60i)T + 59iT^{2} \)
61 \( 1 + (6.37 + 11.0i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-11.8 + 3.17i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.16 - 4.34i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.98 + 9.98i)T - 73iT^{2} \)
79 \( 1 + (-1.34 + 2.33i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.0823 + 0.307i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + (2.68 - 10.0i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-1.02 + 3.83i)T + (-84.0 - 48.5i)T^{2} \)
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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

−13.05234898726485966895476909876, −12.34443381477391253800831585842, −11.42341627589721130453655451852, −10.88841883855551029396264429948, −8.503285116265374558427734049094, −8.094243110090268608863382526672, −6.53490196154291846211244518272, −4.92119381911101215061081483124, −3.58168841235858272505351994959, −1.91938954197979256541738388742, 3.52001748734375843034002917630, 4.66557750396078821159613933064, 5.53698421328843441778742689605, 7.15755179291424844946896758325, 8.145237537244317860592599502222, 9.604383071186482842420944455450, 10.84587713094759731448348927488, 11.51225933336017015570799329793, 13.15596142491587332674548900801, 14.04272156042335486695147202481

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