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Properties

Label 2-117-117.5-c1-0-8
Degree $2$
Conductor $117$
Sign $0.713 + 0.700i$
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  + (0.407 − 1.51i)2-s + (1.33 + 1.10i)3-s + (−0.412 − 0.237i)4-s + (−1.74 + 0.466i)5-s + (2.21 − 1.58i)6-s + (0.556 − 2.07i)7-s + (1.69 − 1.69i)8-s + (0.575 + 2.94i)9-s + 2.83i·10-s + (−2.79 − 0.748i)11-s + (−0.289 − 0.771i)12-s + (0.478 + 3.57i)13-s + (−2.93 − 1.69i)14-s + (−2.84 − 1.29i)15-s + (−2.36 − 4.09i)16-s − 6.33·17-s + ⋯
L(s)  = 1  + (0.287 − 1.07i)2-s + (0.771 + 0.635i)3-s + (−0.206 − 0.118i)4-s + (−0.778 + 0.208i)5-s + (0.905 − 0.646i)6-s + (0.210 − 0.785i)7-s + (0.599 − 0.599i)8-s + (0.191 + 0.981i)9-s + 0.896i·10-s + (−0.842 − 0.225i)11-s + (−0.0834 − 0.222i)12-s + (0.132 + 0.991i)13-s + (−0.783 − 0.452i)14-s + (−0.733 − 0.333i)15-s + (−0.590 − 1.02i)16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.713 + 0.700i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (5, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.713 + 0.700i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30742 - 0.534441i\)
\(L(\frac12)\) \(\approx\) \(1.30742 - 0.534441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.33 - 1.10i)T \)
13 \( 1 + (-0.478 - 3.57i)T \)
good2 \( 1 + (-0.407 + 1.51i)T + (-1.73 - i)T^{2} \)
5 \( 1 + (1.74 - 0.466i)T + (4.33 - 2.5i)T^{2} \)
7 \( 1 + (-0.556 + 2.07i)T + (-6.06 - 3.5i)T^{2} \)
11 \( 1 + (2.79 + 0.748i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 6.33T + 17T^{2} \)
19 \( 1 + (-0.0431 + 0.0431i)T - 19iT^{2} \)
23 \( 1 + (1.17 - 2.02i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-8.31 + 4.80i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.602 - 2.24i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (3.87 + 3.87i)T + 37iT^{2} \)
41 \( 1 + (-2.06 + 0.554i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.40 + 4.27i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-4.53 - 1.21i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 - 3.15iT - 53T^{2} \)
59 \( 1 + (0.734 + 2.74i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (4.45 + 7.71i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.23 + 12.0i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-6.13 - 6.13i)T + 71iT^{2} \)
73 \( 1 + (-0.333 - 0.333i)T + 73iT^{2} \)
79 \( 1 + (-6.64 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.46 - 5.45i)T + (-71.8 - 41.5i)T^{2} \)
89 \( 1 + (4.52 - 4.52i)T - 89iT^{2} \)
97 \( 1 + (13.7 + 3.67i)T + (84.0 + 48.5i)T^{2} \)
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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

−13.60119323382196631620073561049, −12.24792527896517126361533063146, −11.03858870435070693989393185587, −10.69339395829467675304170555033, −9.413933662321959415649752351178, −8.075287219526762026085932819966, −7.05998201893677852076029425552, −4.52134247818335848875267024127, −3.81354764567519320577697332492, −2.38308041231881869072133355006, 2.53669219358594337073738208133, 4.57899159473941756463408613194, 5.99151765062679406972821408264, 7.18345886548515813700559696308, 8.134486229160459867053814555422, 8.705978717258750164544339416611, 10.56062851032966036241755761176, 11.90497766816820817117084221697, 12.89013277243776852243986254054, 13.80216895876508590501288572582

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