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Properties

Label 2-117-117.103-c3-0-31
Degree $2$
Conductor $117$
Sign $0.957 + 0.287i$
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  + (3.17 − 1.83i)2-s + (4.26 + 2.96i)3-s + (2.72 − 4.72i)4-s + (−1.60 − 0.926i)5-s + (18.9 + 1.60i)6-s + (12.1 − 7.01i)7-s + 9.33i·8-s + (9.39 + 25.3i)9-s − 6.80·10-s + (52.3 − 30.2i)11-s + (25.6 − 12.0i)12-s + (−31.9 − 34.3i)13-s + (25.7 − 44.5i)14-s + (−4.09 − 8.71i)15-s + (38.9 + 67.4i)16-s − 82.6·17-s + ⋯
L(s)  = 1  + (1.12 − 0.648i)2-s + (0.820 + 0.571i)3-s + (0.340 − 0.590i)4-s + (−0.143 − 0.0829i)5-s + (1.29 + 0.109i)6-s + (0.656 − 0.378i)7-s + 0.412i·8-s + (0.347 + 0.937i)9-s − 0.215·10-s + (1.43 − 0.828i)11-s + (0.617 − 0.290i)12-s + (−0.681 − 0.732i)13-s + (0.491 − 0.851i)14-s + (−0.0705 − 0.150i)15-s + (0.608 + 1.05i)16-s − 1.17·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.957 + 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.957 + 0.287i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.957 + 0.287i)\)

Particular Values

\(L(2)\) \(\approx\) \(3.58320 - 0.525917i\)
\(L(\frac12)\) \(\approx\) \(3.58320 - 0.525917i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-4.26 - 2.96i)T \)
13 \( 1 + (31.9 + 34.3i)T \)
good2 \( 1 + (-3.17 + 1.83i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (1.60 + 0.926i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-12.1 + 7.01i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-52.3 + 30.2i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + 82.6T + 4.91e3T^{2} \)
19 \( 1 - 66.3iT - 6.85e3T^{2} \)
23 \( 1 + (69.7 - 120. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (148. + 257. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (19.3 + 11.1i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 113. iT - 5.06e4T^{2} \)
41 \( 1 + (402. + 232. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-149. - 258. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-58.4 + 33.7i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 - 334.T + 1.48e5T^{2} \)
59 \( 1 + (-346. - 200. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-347. - 602. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-72.8 - 42.0i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + 877. iT - 3.57e5T^{2} \)
73 \( 1 - 355. iT - 3.89e5T^{2} \)
79 \( 1 + (-186. - 322. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-73.8 + 42.6i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 + 1.23e3iT - 7.04e5T^{2} \)
97 \( 1 + (-0.0892 + 0.0515i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.29276072803722738167474891309, −11.92395613625545933364649401085, −11.21907339082740993208330344243, −10.03164656683375203565877316161, −8.714129601189228196317160472150, −7.73500665441527560560187593109, −5.75059795321871974825940616942, −4.31916253003684643124204095213, −3.68301011093967604055408961657, −2.07285054369034860125019623954, 1.95141450894396441211003478577, 3.84292723003217672713420757153, 4.87919282020541831480674643216, 6.67640378993121173205758607343, 7.08923203557079314126329302842, 8.689710586526978716055039055165, 9.587992771230976453675354253498, 11.56670671503117751528557548286, 12.37364412405542295953382867878, 13.32434243892102724809730441426

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