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Properties

Label 2-117-13.10-c3-0-13
Degree $2$
Conductor $117$
Sign $-0.0639 + 0.997i$
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  + (4.24 − 2.45i)2-s + (8.02 − 13.8i)4-s − 5.61i·5-s + (2.78 + 1.60i)7-s − 39.4i·8-s + (−13.7 − 23.8i)10-s + (37.7 − 21.8i)11-s + (−46.8 − 2.39i)13-s + 15.7·14-s + (−32.5 − 56.3i)16-s + (−32.9 + 57.0i)17-s + (−2.65 − 1.53i)19-s + (−78.0 − 45.0i)20-s + (107. − 185. i)22-s + (81.8 + 141. i)23-s + ⋯
L(s)  = 1  + (1.50 − 0.866i)2-s + (1.00 − 1.73i)4-s − 0.502i·5-s + (0.150 + 0.0867i)7-s − 1.74i·8-s + (−0.435 − 0.753i)10-s + (1.03 − 0.598i)11-s + (−0.998 − 0.0511i)13-s + 0.300·14-s + (−0.508 − 0.880i)16-s + (−0.469 + 0.813i)17-s + (−0.0320 − 0.0184i)19-s + (−0.872 − 0.503i)20-s + (1.03 − 1.79i)22-s + (0.742 + 1.28i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0639 + 0.997i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0639 + 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.0639 + 0.997i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ -0.0639 + 0.997i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.43884 - 2.60017i\)
\(L(\frac12)\) \(\approx\) \(2.43884 - 2.60017i\)
\(L(\frac{5}{2})\) 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 \( 1 + (46.8 + 2.39i)T \)
good2 \( 1 + (-4.24 + 2.45i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + 5.61iT - 125T^{2} \)
7 \( 1 + (-2.78 - 1.60i)T + (171.5 + 297. i)T^{2} \)
11 \( 1 + (-37.7 + 21.8i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (32.9 - 57.0i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (2.65 + 1.53i)T + (3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-81.8 - 141. i)T + (-6.08e3 + 1.05e4i)T^{2} \)
29 \( 1 + (-77.7 - 134. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + 240. iT - 2.97e4T^{2} \)
37 \( 1 + (275. - 158. i)T + (2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (110. - 63.9i)T + (3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (189. - 328. i)T + (-3.97e4 - 6.88e4i)T^{2} \)
47 \( 1 - 79.4iT - 1.03e5T^{2} \)
53 \( 1 - 571.T + 1.48e5T^{2} \)
59 \( 1 + (-91.3 - 52.7i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (-362. + 627. i)T + (-1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (261. - 151. i)T + (1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (868. + 501. i)T + (1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 277. iT - 3.89e5T^{2} \)
79 \( 1 + 959.T + 4.93e5T^{2} \)
83 \( 1 + 410. iT - 5.71e5T^{2} \)
89 \( 1 + (195. - 113. i)T + (3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (769. + 444. i)T + (4.56e5 + 7.90e5i)T^{2} \)
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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

−12.81856542315849053775304822158, −11.87460058071713567587143262195, −11.20040810026762205177416646610, −9.942510119945734268363085112612, −8.635994128388267560889015195730, −6.77167139805206462345581519230, −5.49449692128833916774991332183, −4.47888878322484391034968081918, −3.21733344005454700841603650009, −1.52572572456582849386574327776, 2.72426351842768423030022232885, 4.23782240738996043264900071987, 5.19154250992170121251658524700, 6.84025732071668995291924497898, 7.03892999466500037161566926328, 8.781712638957524793776426371956, 10.36170268817411569868048498087, 11.81547140705099001437256869528, 12.39843523955105998986294838091, 13.60326818579801404744043786241

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