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Properties

Label 2-117-13.4-c3-0-7
Degree $2$
Conductor $117$
Sign $-0.995 + 0.0972i$
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  + (−4.36 − 2.52i)2-s + (8.72 + 15.1i)4-s + 20.1i·5-s + (−13.3 + 7.71i)7-s − 47.7i·8-s + (50.7 − 87.9i)10-s + (−23.3 − 13.4i)11-s + (−3.96 − 46.7i)13-s + 77.8·14-s + (−50.5 + 87.5i)16-s + (−11.6 − 20.1i)17-s + (−39.0 + 22.5i)19-s + (−304. + 175. i)20-s + (67.9 + 117. i)22-s + (71.0 − 122. i)23-s + ⋯
L(s)  = 1  + (−1.54 − 0.891i)2-s + (1.09 + 1.88i)4-s + 1.79i·5-s + (−0.721 + 0.416i)7-s − 2.10i·8-s + (1.60 − 2.77i)10-s + (−0.639 − 0.369i)11-s + (−0.0845 − 0.996i)13-s + 1.48·14-s + (−0.789 + 1.36i)16-s + (−0.165 − 0.287i)17-s + (−0.471 + 0.272i)19-s + (−3.40 + 1.96i)20-s + (0.658 + 1.14i)22-s + (0.643 − 1.11i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0972i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.995 + 0.0972i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.995 + 0.0972i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.000871256 - 0.0178715i\)
\(L(\frac12)\) \(\approx\) \(0.000871256 - 0.0178715i\)
\(L(\frac{5}{2})\) 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 \( 1 + (3.96 + 46.7i)T \)
good2 \( 1 + (4.36 + 2.52i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 20.1iT - 125T^{2} \)
7 \( 1 + (13.3 - 7.71i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (23.3 + 13.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (11.6 + 20.1i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (39.0 - 22.5i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-71.0 + 122. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-1.14 + 1.98i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 37.7iT - 2.97e4T^{2} \)
37 \( 1 + (-271. - 156. i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (5.08 + 2.93i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (180. + 312. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 209. iT - 1.03e5T^{2} \)
53 \( 1 + 276.T + 1.48e5T^{2} \)
59 \( 1 + (470. - 271. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (102. + 178. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (426. + 246. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (716. - 413. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 66.1iT - 3.89e5T^{2} \)
79 \( 1 - 317.T + 4.93e5T^{2} \)
83 \( 1 + 141. iT - 5.71e5T^{2} \)
89 \( 1 + (555. + 320. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (965. - 557. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−12.13621662757797663447239249028, −10.97101268083549110278987555150, −10.49777129560659807165619103453, −9.679950974726834429511285993654, −8.353925849274682711943767030847, −7.33292992908656026291625630373, −6.23807166538977547153195092369, −3.16712023508896116252932735410, −2.55890551379439691615600883594, −0.01577540684956763923668756034, 1.46213479535753607095760453552, 4.64120854859964952474978571004, 6.00852343248621742721232314030, 7.28693815889179826225210507390, 8.298934610104102665035265505314, 9.262413628034002286553520814421, 9.740248508611170408315025499927, 11.15280541387272579814636363507, 12.60588241498336083035088021707, 13.47101157193283513651037975277

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