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Properties

Label 2-117-13.4-c3-0-9
Degree $2$
Conductor $117$
Sign $-0.0302 - 0.999i$
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  + (4.64 + 2.68i)2-s + (10.3 + 17.9i)4-s + 2.69i·5-s + (13.1 − 7.60i)7-s + 68.5i·8-s + (−7.23 + 12.5i)10-s + (−57.9 − 33.4i)11-s + (46.8 − 0.818i)13-s + 81.5·14-s + (−100. + 174. i)16-s + (−2.08 − 3.60i)17-s + (−22.5 + 13.0i)19-s + (−48.5 + 28.0i)20-s + (−179. − 310. i)22-s + (−23.6 + 40.9i)23-s + ⋯
L(s)  = 1  + (1.64 + 0.948i)2-s + (1.29 + 2.24i)4-s + 0.241i·5-s + (0.710 − 0.410i)7-s + 3.03i·8-s + (−0.228 + 0.396i)10-s + (−1.58 − 0.916i)11-s + (0.999 − 0.0174i)13-s + 1.55·14-s + (−1.57 + 2.72i)16-s + (−0.0297 − 0.0514i)17-s + (−0.272 + 0.157i)19-s + (−0.542 + 0.313i)20-s + (−1.73 − 3.01i)22-s + (−0.214 + 0.371i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0302 - 0.999i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0302 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.0302 - 0.999i$
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: \((2,\ 117,\ (\ :3/2),\ -0.0302 - 0.999i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.72845 + 2.81237i\)
\(L(\frac12)\) \(\approx\) \(2.72845 + 2.81237i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-46.8 + 0.818i)T \)
good2 \( 1 + (-4.64 - 2.68i)T + (4 + 6.92i)T^{2} \)
5 \( 1 - 2.69iT - 125T^{2} \)
7 \( 1 + (-13.1 + 7.60i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (57.9 + 33.4i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (2.08 + 3.60i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (22.5 - 13.0i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (23.6 - 40.9i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-128. + 222. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 206. iT - 2.97e4T^{2} \)
37 \( 1 + (152. + 87.8i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (135. + 78.2i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-25.9 - 45.0i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 354. iT - 1.03e5T^{2} \)
53 \( 1 - 10.4T + 1.48e5T^{2} \)
59 \( 1 + (385. - 222. i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (59.8 + 103. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (19.4 + 11.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (-246. + 142. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 740. iT - 3.89e5T^{2} \)
79 \( 1 + 547.T + 4.93e5T^{2} \)
83 \( 1 - 603. iT - 5.71e5T^{2} \)
89 \( 1 + (-186. - 107. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (1.25e3 - 723. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.59556910577069062552890748997, −12.75224628101544512526978690335, −11.43757286861099173677844315803, −10.70711679018690728423635917579, −8.309694019940159514127284126234, −7.68308448271730877689898777270, −6.29772320352987879163312414844, −5.37255695825855468228921593401, −4.16227157312794043988955167185, −2.80867258560633203765333819868, 1.70184744392338185862909948722, 3.06037700850888770386932575262, 4.71642295409264241935987930193, 5.29875564616228825630457700803, 6.78320376820159424218097675096, 8.533874287233706179152610079274, 10.33058961562393455705206248468, 10.84993443059603793266686568004, 12.08489990115251001505804189863, 12.73747972457961050147397419258

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