Properties

Label 2-117-13.3-c3-0-5
Degree $2$
Conductor $117$
Sign $0.966 + 0.255i$
Analytic cond. $6.90322$
Root an. cond. $2.62739$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−2.28 − 3.95i)2-s + (−6.40 + 11.0i)4-s − 2.80·5-s + (−4.78 + 8.28i)7-s + 21.9·8-s + (6.40 + 11.0i)10-s + (19.7 + 34.1i)11-s + (40.5 + 23.5i)13-s + 43.6·14-s + (1.21 + 2.09i)16-s + (1.00 − 1.74i)17-s + (30.0 − 52.1i)19-s + (17.9 − 31.1i)20-s + (89.9 − 155. i)22-s + (2.23 + 3.87i)23-s + ⋯
L(s)  = 1  + (−0.806 − 1.39i)2-s + (−0.800 + 1.38i)4-s − 0.251·5-s + (−0.258 + 0.447i)7-s + 0.969·8-s + (0.202 + 0.350i)10-s + (0.540 + 0.935i)11-s + (0.864 + 0.502i)13-s + 0.832·14-s + (0.0189 + 0.0327i)16-s + (0.0143 − 0.0248i)17-s + (0.363 − 0.629i)19-s + (0.201 − 0.348i)20-s + (0.871 − 1.50i)22-s + (0.0202 + 0.0350i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.966 + 0.255i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.966 + 0.255i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (55, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.966 + 0.255i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.814604 - 0.105743i\)
\(L(\frac12)\) \(\approx\) \(0.814604 - 0.105743i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-40.5 - 23.5i)T \)
good2 \( 1 + (2.28 + 3.95i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 + 2.80T + 125T^{2} \)
7 \( 1 + (4.78 - 8.28i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.7 - 34.1i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-1.00 + 1.74i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-30.0 + 52.1i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-2.23 - 3.87i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-70.3 - 121. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 136.T + 2.97e4T^{2} \)
37 \( 1 + (-92.8 - 160. i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-155. - 268. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (213. - 370. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 258.T + 1.03e5T^{2} \)
53 \( 1 + 612.T + 1.48e5T^{2} \)
59 \( 1 + (258. - 448. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-80.6 + 139. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-24.9 - 43.2i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-139. + 242. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 467.T + 3.89e5T^{2} \)
79 \( 1 - 37.5T + 4.93e5T^{2} \)
83 \( 1 - 76.1T + 5.71e5T^{2} \)
89 \( 1 + (-101. - 175. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-587. + 1.01e3i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.55972882097457973688208329734, −11.78783925764213768551444097734, −11.02509755650489482213384277138, −9.775893215389128979278685011546, −9.149672471637807427166131400978, −8.001110152086606539553685311595, −6.41561199469595640826864858314, −4.35969519495631296521279912842, −2.93772806626954416843494108998, −1.39681208255563442820084964064, 0.64953837019483212290604727153, 3.72347384732275396379261576408, 5.65491551289112755107606978597, 6.49964946354975196595654909878, 7.73845250006464880020502962043, 8.498290855953406932059923049203, 9.587323920624907293278998328535, 10.69942068614372338393421520937, 11.99881716856588620138698979167, 13.57113712160110848401270387524

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