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Properties

Label 2-117-13.10-c7-0-13
Degree $2$
Conductor $117$
Sign $0.443 - 0.896i$
Analytic cond. $36.5490$
Root an. cond. $6.04558$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−15.0 + 8.67i)2-s + (86.6 − 150. i)4-s − 301. i·5-s + (835. + 482. i)7-s + 787. i·8-s + (2.61e3 + 4.53e3i)10-s + (−6.04e3 + 3.49e3i)11-s + (−7.14e3 − 3.42e3i)13-s − 1.67e4·14-s + (4.25e3 + 7.37e3i)16-s + (2.52e3 − 4.37e3i)17-s + (−7.72e3 − 4.45e3i)19-s + (−4.52e4 − 2.61e4i)20-s + (6.06e4 − 1.04e5i)22-s + (4.66e4 + 8.07e4i)23-s + ⋯
L(s)  = 1  + (−1.32 + 0.767i)2-s + (0.677 − 1.17i)4-s − 1.07i·5-s + (0.920 + 0.531i)7-s + 0.543i·8-s + (0.827 + 1.43i)10-s + (−1.36 + 0.790i)11-s + (−0.901 − 0.432i)13-s − 1.63·14-s + (0.259 + 0.450i)16-s + (0.124 − 0.215i)17-s + (−0.258 − 0.149i)19-s + (−1.26 − 0.730i)20-s + (1.21 − 2.10i)22-s + (0.798 + 1.38i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.443 - 0.896i$
Analytic conductor: \(36.5490\)
Root analytic conductor: \(6.04558\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :7/2),\ 0.443 - 0.896i)\)

Particular Values

\(L(4)\) \(\approx\) \(0.7725603998\)
\(L(\frac12)\) \(\approx\) \(0.7725603998\)
\(L(\frac{9}{2})\) 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 \( 1 + (7.14e3 + 3.42e3i)T \)
good2 \( 1 + (15.0 - 8.67i)T + (64 - 110. i)T^{2} \)
5 \( 1 + 301. iT - 7.81e4T^{2} \)
7 \( 1 + (-835. - 482. i)T + (4.11e5 + 7.13e5i)T^{2} \)
11 \( 1 + (6.04e3 - 3.49e3i)T + (9.74e6 - 1.68e7i)T^{2} \)
17 \( 1 + (-2.52e3 + 4.37e3i)T + (-2.05e8 - 3.55e8i)T^{2} \)
19 \( 1 + (7.72e3 + 4.45e3i)T + (4.46e8 + 7.74e8i)T^{2} \)
23 \( 1 + (-4.66e4 - 8.07e4i)T + (-1.70e9 + 2.94e9i)T^{2} \)
29 \( 1 + (5.96e4 + 1.03e5i)T + (-8.62e9 + 1.49e10i)T^{2} \)
31 \( 1 + 1.72e5iT - 2.75e10T^{2} \)
37 \( 1 + (7.43e4 - 4.29e4i)T + (4.74e10 - 8.22e10i)T^{2} \)
41 \( 1 + (-5.60e5 + 3.23e5i)T + (9.73e10 - 1.68e11i)T^{2} \)
43 \( 1 + (8.05e4 - 1.39e5i)T + (-1.35e11 - 2.35e11i)T^{2} \)
47 \( 1 - 1.35e6iT - 5.06e11T^{2} \)
53 \( 1 - 2.20e5T + 1.17e12T^{2} \)
59 \( 1 + (-9.69e5 - 5.59e5i)T + (1.24e12 + 2.15e12i)T^{2} \)
61 \( 1 + (8.68e5 - 1.50e6i)T + (-1.57e12 - 2.72e12i)T^{2} \)
67 \( 1 + (-1.62e6 + 9.39e5i)T + (3.03e12 - 5.24e12i)T^{2} \)
71 \( 1 + (1.03e6 + 5.95e5i)T + (4.54e12 + 7.87e12i)T^{2} \)
73 \( 1 + 2.09e6iT - 1.10e13T^{2} \)
79 \( 1 - 7.92e6T + 1.92e13T^{2} \)
83 \( 1 - 7.81e6iT - 2.71e13T^{2} \)
89 \( 1 + (-3.61e6 + 2.08e6i)T + (2.21e13 - 3.83e13i)T^{2} \)
97 \( 1 + (-9.89e6 - 5.71e6i)T + (4.03e13 + 6.99e13i)T^{2} \)
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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

−12.36237155605692432609159965959, −11.01591596307899479044986250078, −9.786859408182000202495588897561, −9.064292773682779332895538446445, −7.901137008160610437305820200185, −7.52554747126147021214989348721, −5.62403259167258043997084242965, −4.78236477628515284158433577375, −2.17054670227110549388622134642, −0.76995376153009445179314661003, 0.52477593663382990677737761606, 2.08921038504704993992445898833, 3.08538036367748537968423108343, 5.05116215961995539588620430890, 6.95689435560008193379620220100, 7.86873551896108319605614874955, 8.785032338606794450074941333560, 10.28943592632131354627039836026, 10.67409617499673138826788228478, 11.38709185653060237868973486059

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