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Properties

Label 2-117-13.4-c3-0-4
Degree $2$
Conductor $117$
Sign $-0.449 - 0.893i$
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  + (3.57 + 2.06i)2-s + (4.50 + 7.79i)4-s + 13.4i·5-s + (−27.2 + 15.7i)7-s + 4.12i·8-s + (−27.7 + 47.9i)10-s + (35.0 + 20.2i)11-s + (42.1 − 20.5i)13-s − 129.·14-s + (27.4 − 47.6i)16-s + (21.5 + 37.3i)17-s + (23.3 − 13.4i)19-s + (−104. + 60.4i)20-s + (83.4 + 144. i)22-s + (9.50 − 16.4i)23-s + ⋯
L(s)  = 1  + (1.26 + 0.728i)2-s + (0.562 + 0.974i)4-s + 1.20i·5-s + (−1.46 + 0.848i)7-s + 0.182i·8-s + (−0.876 + 1.51i)10-s + (0.961 + 0.555i)11-s + (0.898 − 0.438i)13-s − 2.47·14-s + (0.429 − 0.744i)16-s + (0.307 + 0.533i)17-s + (0.282 − 0.162i)19-s + (−1.17 + 0.676i)20-s + (0.809 + 1.40i)22-s + (0.0861 − 0.149i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.449 - 0.893i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.449 - 0.893i$
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: \((2,\ 117,\ (\ :3/2),\ -0.449 - 0.893i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.43552 + 2.32941i\)
\(L(\frac12)\) \(\approx\) \(1.43552 + 2.32941i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-42.1 + 20.5i)T \)
good2 \( 1 + (-3.57 - 2.06i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 13.4iT - 125T^{2} \)
7 \( 1 + (27.2 - 15.7i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-35.0 - 20.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-21.5 - 37.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-23.3 + 13.4i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-9.50 + 16.4i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (77.0 - 133. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 308. iT - 2.97e4T^{2} \)
37 \( 1 + (37.6 + 21.7i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (41.4 + 23.9i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-171. - 296. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 133. iT - 1.03e5T^{2} \)
53 \( 1 - 438.T + 1.48e5T^{2} \)
59 \( 1 + (511. - 295. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-270. - 468. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (199. + 115. i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-389. + 224. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 + 389. iT - 3.89e5T^{2} \)
79 \( 1 + 897.T + 4.93e5T^{2} \)
83 \( 1 + 1.30e3iT - 5.71e5T^{2} \)
89 \( 1 + (801. + 462. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.35e3 + 780. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.41615353115974137110208973891, −12.70115853119304043945714803876, −11.68454936838461294200767599822, −10.25970098390000358733756609277, −9.173980189741531548638152263738, −7.31436531658410065276883560417, −6.38159540674058900720987976037, −5.82814022782418992487926580641, −3.88012526361861284910860452040, −2.96043130279848830693760204051, 1.10253552640984751063628396417, 3.36938284121497362544289211991, 4.16455480234032110363751338535, 5.57129745496487516330860916589, 6.73228341635812586772337739859, 8.634456868856747837392145629244, 9.658415912217608923318077272023, 10.96365126823424406822424929072, 12.05812642774135165128777164712, 12.74568985587549111840258593923

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