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Properties

Label 2-117-13.3-c3-0-7
Degree $2$
Conductor $117$
Sign $-0.927 - 0.372i$
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  + (2.66 + 4.61i)2-s + (−10.2 + 17.6i)4-s + 16.4·5-s + (−4.83 + 8.38i)7-s − 66.1·8-s + (43.7 + 75.7i)10-s + (13.7 + 23.8i)11-s + (−37.3 − 28.3i)13-s − 51.5·14-s + (−94.6 − 164. i)16-s + (53.9 − 93.4i)17-s + (1.12 − 1.94i)19-s + (−167. + 290. i)20-s + (−73.5 + 127. i)22-s + (20.9 + 36.2i)23-s + ⋯
L(s)  = 1  + (0.942 + 1.63i)2-s + (−1.27 + 2.20i)4-s + 1.46·5-s + (−0.261 + 0.452i)7-s − 2.92·8-s + (1.38 + 2.39i)10-s + (0.378 + 0.654i)11-s + (−0.795 − 0.605i)13-s − 0.984·14-s + (−1.47 − 2.56i)16-s + (0.769 − 1.33i)17-s + (0.0135 − 0.0234i)19-s + (−1.87 + 3.24i)20-s + (−0.712 + 1.23i)22-s + (0.189 + 0.328i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.927 - 0.372i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.927 - 0.372i$
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: \((2,\ 117,\ (\ :3/2),\ -0.927 - 0.372i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.524368 + 2.71066i\)
\(L(\frac12)\) \(\approx\) \(0.524368 + 2.71066i\)
\(L(\frac{5}{2})\) 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 \( 1 + (37.3 + 28.3i)T \)
good2 \( 1 + (-2.66 - 4.61i)T + (-4 + 6.92i)T^{2} \)
5 \( 1 - 16.4T + 125T^{2} \)
7 \( 1 + (4.83 - 8.38i)T + (-171.5 - 297. i)T^{2} \)
11 \( 1 + (-13.7 - 23.8i)T + (-665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-53.9 + 93.4i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-1.12 + 1.94i)T + (-3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-20.9 - 36.2i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-30.8 - 53.3i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 - 191.T + 2.97e4T^{2} \)
37 \( 1 + (49.2 + 85.2i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (15.3 + 26.6i)T + (-3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (119. - 206. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 511.T + 1.03e5T^{2} \)
53 \( 1 + 492.T + 1.48e5T^{2} \)
59 \( 1 + (-242. + 419. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-222. + 384. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (95.0 + 164. i)T + (-1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-242. + 419. i)T + (-1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 957.T + 3.89e5T^{2} \)
79 \( 1 + 375.T + 4.93e5T^{2} \)
83 \( 1 - 715.T + 5.71e5T^{2} \)
89 \( 1 + (519. + 899. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (32.7 - 56.7i)T + (-4.56e5 - 7.90e5i)T^{2} \)
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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

−13.84879358309486068191935851718, −12.84639231904810104795267286930, −12.09485714004995809979139718101, −9.870454503457012128269643062331, −9.126700473843979618851975934167, −7.64494350349814081131367811703, −6.62714425677483464191453797196, −5.62236823699782108444508691785, −4.85135499634109144922123071692, −2.86043986996450382314465335678, 1.28917857556260494426245516694, 2.58702191379933373998341470933, 4.03733320968947086187304431528, 5.43101326060599764639928711255, 6.38535143182214809227397922134, 8.906536242664054708212185028862, 10.03803453567713998745262981825, 10.37391482937617133421084162411, 11.71454243191778962878220788967, 12.63252287750944153456704706012

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