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Properties

Label 2-117-13.12-c1-0-1
Degree $2$
Conductor $117$
Sign $-0.277 - 0.960i$
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  + 1.73i·2-s − 0.999·4-s + 3.46i·7-s + 1.73i·8-s − 3.46i·11-s + (−1 − 3.46i)13-s − 5.99·14-s − 5·16-s + 6·17-s − 3.46i·19-s + 5.99·22-s + 5·25-s + (5.99 − 1.73i)26-s − 3.46i·28-s − 6·29-s + ⋯
L(s)  = 1  + 1.22i·2-s − 0.499·4-s + 1.30i·7-s + 0.612i·8-s − 1.04i·11-s + (−0.277 − 0.960i)13-s − 1.60·14-s − 1.25·16-s + 1.45·17-s − 0.794i·19-s + 1.27·22-s + 25-s + (1.17 − 0.339i)26-s − 0.654i·28-s − 1.11·29-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.277 - 0.960i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.277 - 0.960i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (64, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ -0.277 - 0.960i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.656312 + 0.872573i\)
\(L(\frac12)\) \(\approx\) \(0.656312 + 0.872573i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 + 3.46i)T \)
good2 \( 1 - 1.73iT - 2T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 - 3.46iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
17 \( 1 - 6T + 17T^{2} \)
19 \( 1 + 3.46iT - 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + 6T + 29T^{2} \)
31 \( 1 - 3.46iT - 31T^{2} \)
37 \( 1 + 6.92iT - 37T^{2} \)
41 \( 1 + 6.92iT - 41T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 6T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 - 3.46iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + 3.46iT - 83T^{2} \)
89 \( 1 + 6.92iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−14.25796646071668065415399511013, −12.92870767676208291067046516078, −11.87358826855113987605804394838, −10.74876269550658365604509198493, −9.132630457026905491932237417534, −8.340846955215312984274751734418, −7.26576752323852867591516307774, −5.82159304803056284359610985675, −5.34017448720661284081589497718, −2.91134212354398266583078098641, 1.57686584716747167281817741029, 3.47018727620373522194747060402, 4.60565980587590345138887976369, 6.72093424702019735750393022276, 7.71327862265261631414445850877, 9.615125411710301297732032956462, 10.10812231428556152192195977023, 11.16719593149364785823632598205, 12.12513433543839575821358493524, 12.94653876664632992175973565717

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