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Properties

Label 2-117-13.3-c3-0-15
Degree $2$
Conductor $117$
Sign $-0.969 - 0.243i$
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  + (−2.11 − 3.65i)2-s + (−4.92 + 8.53i)4-s + 5.85·5-s + (12.0 − 20.8i)7-s + 7.85·8-s + (−12.3 − 21.4i)10-s + (−16.9 − 29.3i)11-s + (−40.8 − 23.0i)13-s − 101.·14-s + (22.8 + 39.5i)16-s + (−24.6 + 42.7i)17-s + (38.4 − 66.5i)19-s + (−28.8 + 50.0i)20-s + (−71.6 + 124. i)22-s + (3.14 + 5.44i)23-s + ⋯
L(s)  = 1  + (−0.747 − 1.29i)2-s + (−0.616 + 1.06i)4-s + 0.524·5-s + (0.651 − 1.12i)7-s + 0.347·8-s + (−0.391 − 0.678i)10-s + (−0.464 − 0.804i)11-s + (−0.870 − 0.492i)13-s − 1.94·14-s + (0.356 + 0.617i)16-s + (−0.352 + 0.610i)17-s + (0.463 − 0.803i)19-s + (−0.322 + 0.559i)20-s + (−0.693 + 1.20i)22-s + (0.0285 + 0.0493i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.243i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.969 - 0.243i$
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: \((2,\ 117,\ (\ :3/2),\ -0.969 - 0.243i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.104487 + 0.843887i\)
\(L(\frac12)\) \(\approx\) \(0.104487 + 0.843887i\)
\(L(\frac{5}{2})\) 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 \( 1 + (40.8 + 23.0i)T \)
good2 \( 1 + (2.11 + 3.65i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 5.85T + 125T^{2} \)
7 \( 1 + (-12.0 + 20.8i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (16.9 + 29.3i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (24.6 - 42.7i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-38.4 + 66.5i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-3.14 - 5.44i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-50.4 - 87.4i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 307.T + 2.97e4T^{2} \)
37 \( 1 + (-38.0 - 65.8i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (257. + 445. i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-134. + 232. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 460.T + 1.03e5T^{2} \)
53 \( 1 + 67.8T + 1.48e5T^{2} \)
59 \( 1 + (-12.6 + 21.8i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-294. + 509. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-502. - 869. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-447. + 775. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 968.T + 3.89e5T^{2} \)
79 \( 1 + 119.T + 4.93e5T^{2} \)
83 \( 1 + 480.T + 5.71e5T^{2} \)
89 \( 1 + (-542. - 940. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-8.32 + 14.4i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−12.29466937899417656627437145469, −11.00132474468167384849069499221, −10.62504287534274049095315839681, −9.581788897917566569541329531935, −8.465877567638460551237338034450, −7.27686625716906417275188226292, −5.40535381965344023482461623010, −3.67289092598623763170501244654, −2.11963184329124977292573159610, −0.56199939252510223446235271449, 2.21095331643075191704044955465, 4.98879656193582297184248004887, 5.87374371350409643507881370209, 7.19059002260195457552951196308, 8.088707153541948664887162655182, 9.259127196278467441088999407209, 9.859803648698570586038619432016, 11.58350697207547847830196767646, 12.55832412599445131429316302418, 14.08728217329759968417180797178

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