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Properties

Label 2-117-13.10-c1-0-0
Degree $2$
Conductor $117$
Sign $0.252 - 0.967i$
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 + 1.73i)4-s + 3.46i·5-s + (−1.5 − 0.866i)7-s + (3 − 1.73i)11-s + (3.5 + 0.866i)13-s + (−1.99 − 3.46i)16-s + (3 + 1.73i)19-s + (−5.99 − 3.46i)20-s + (−3 − 5.19i)23-s − 6.99·25-s + (3 − 1.73i)28-s + (3 + 5.19i)29-s − 1.73i·31-s + (2.99 − 5.19i)35-s + (6 − 3.46i)41-s + ⋯
L(s)  = 1  + (−0.5 + 0.866i)4-s + 1.54i·5-s + (−0.566 − 0.327i)7-s + (0.904 − 0.522i)11-s + (0.970 + 0.240i)13-s + (−0.499 − 0.866i)16-s + (0.688 + 0.397i)19-s + (−1.34 − 0.774i)20-s + (−0.625 − 1.08i)23-s − 1.39·25-s + (0.566 − 0.327i)28-s + (0.557 + 0.964i)29-s − 0.311i·31-s + (0.507 − 0.878i)35-s + (0.937 − 0.541i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(0.934249\)
Root analytic conductor: \(0.966565\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1/2),\ 0.252 - 0.967i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.741647 + 0.572873i\)
\(L(\frac12)\) \(\approx\) \(0.741647 + 0.572873i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.5 - 0.866i)T \)
good2 \( 1 + (1 - 1.73i)T^{2} \)
5 \( 1 - 3.46iT - 5T^{2} \)
7 \( 1 + (1.5 + 0.866i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-3 + 1.73i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3 - 1.73i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (3 + 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (-6 + 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.5 + 0.866i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.46iT - 47T^{2} \)
53 \( 1 + 12T + 53T^{2} \)
59 \( 1 + (-3 - 1.73i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-7.5 + 4.33i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-9 - 5.19i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.73iT - 73T^{2} \)
79 \( 1 + 11T + 79T^{2} \)
83 \( 1 + 13.8iT - 83T^{2} \)
89 \( 1 + (6 - 3.46i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (4.5 + 2.59i)T + (48.5 + 84.0i)T^{2} \)
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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

−14.04841022942684093750473571811, −12.76560893809697189766208200774, −11.62132858519400011466854118130, −10.71461154115753083249230894512, −9.539018328055378026618697972994, −8.330447422381186126035980313518, −7.06727364404547877490573285254, −6.22918150692400218279346685722, −3.95264982477068541653487925411, −3.09036504556615477107757497699, 1.24814228878813644848669456513, 4.09655384127961639852717399175, 5.28059496278388864830516026578, 6.30168178556469811099091670290, 8.215411476473491959655593083959, 9.294476075060665362368958814769, 9.702645435345651152911373397047, 11.35842360822776695384530967690, 12.45543798647023413397316385650, 13.30646623313535601246384551851

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