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Properties

Label 2-117-13.9-c3-0-9
Degree $2$
Conductor $117$
Sign $0.525 + 0.850i$
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  + (1.84 − 3.19i)2-s + (−2.81 − 4.87i)4-s + 18.7·5-s + (12.0 + 20.9i)7-s + 8.74·8-s + (34.6 − 59.9i)10-s + (−24.6 + 42.6i)11-s + (−34.3 − 31.9i)13-s + 89.1·14-s + (38.6 − 66.9i)16-s + (−32.6 − 56.5i)17-s + (−54.9 − 95.2i)19-s + (−52.8 − 91.4i)20-s + (90.9 + 157. i)22-s + (−41.6 + 72.1i)23-s + ⋯
L(s)  = 1  + (0.652 − 1.13i)2-s + (−0.351 − 0.609i)4-s + 1.67·5-s + (0.651 + 1.12i)7-s + 0.386·8-s + (1.09 − 1.89i)10-s + (−0.675 + 1.16i)11-s + (−0.732 − 0.681i)13-s + 1.70·14-s + (0.604 − 1.04i)16-s + (−0.465 − 0.806i)17-s + (−0.663 − 1.14i)19-s + (−0.590 − 1.02i)20-s + (0.881 + 1.52i)22-s + (−0.377 + 0.653i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.525 + 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.525 + 0.850i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.525 + 0.850i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.55474 - 1.42520i\)
\(L(\frac12)\) \(\approx\) \(2.55474 - 1.42520i\)
\(L(\frac{5}{2})\) 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 \( 1 + (34.3 + 31.9i)T \)
good2 \( 1 + (-1.84 + 3.19i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 18.7T + 125T^{2} \)
7 \( 1 + (-12.0 - 20.9i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (24.6 - 42.6i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (32.6 + 56.5i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (54.9 + 95.2i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (41.6 - 72.1i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-2.49 + 4.32i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 255.T + 2.97e4T^{2} \)
37 \( 1 + (46.8 - 81.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-33.9 + 58.8i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (71.2 + 123. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 379.T + 1.03e5T^{2} \)
53 \( 1 - 389.T + 1.48e5T^{2} \)
59 \( 1 + (-66.9 - 115. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (310. + 537. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (59.5 - 103. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (-180. - 312. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 + 748.T + 3.89e5T^{2} \)
79 \( 1 - 514.T + 4.93e5T^{2} \)
83 \( 1 + 260.T + 5.71e5T^{2} \)
89 \( 1 + (416. - 721. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (-740. - 1.28e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.86023467379337051103149716576, −12.04136323179511858493449778142, −10.85142498682870246782348064852, −9.948948636741992091954725129870, −9.055006593594085882170968127831, −7.28272516956185936214279043907, −5.47532051889228650872598274762, −4.88502981126234637370316108334, −2.51144294686675963981910488213, −2.05351089065379083337985209717, 1.81911609205266016828681485425, 4.24509402682492028311534311846, 5.52398175123917375608256682899, 6.27307068937365905061592970086, 7.45117183776166981001962957340, 8.670331347757884422740198079649, 10.27144502850884541843538098873, 10.77987963088653241228642465105, 12.81146805333651766073772368772, 13.62010052459184037574715829962

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