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Properties

Label 2-117-13.5-c4-0-19
Degree $2$
Conductor $117$
Sign $-0.636 + 0.771i$
Analytic cond. $12.0942$
Root an. cond. $3.47768$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.32 − 4.32i)2-s − 21.4i·4-s + (26.0 − 26.0i)5-s + (−16.9 − 16.9i)7-s + (−23.7 − 23.7i)8-s − 225. i·10-s + (69.5 + 69.5i)11-s + (−93.5 − 140. i)13-s − 146.·14-s + 138.·16-s − 166. i·17-s + (−468. + 468. i)19-s + (−560. − 560. i)20-s + 602.·22-s − 388. i·23-s + ⋯
L(s)  = 1  + (1.08 − 1.08i)2-s − 1.34i·4-s + (1.04 − 1.04i)5-s + (−0.346 − 0.346i)7-s + (−0.370 − 0.370i)8-s − 2.25i·10-s + (0.574 + 0.574i)11-s + (−0.553 − 0.832i)13-s − 0.749·14-s + 0.539·16-s − 0.576i·17-s + (−1.29 + 1.29i)19-s + (−1.40 − 1.40i)20-s + 1.24·22-s − 0.735i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.636 + 0.771i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $-0.636 + 0.771i$
Analytic conductor: \(12.0942\)
Root analytic conductor: \(3.47768\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :2),\ -0.636 + 0.771i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.48728 - 3.15548i\)
\(L(\frac12)\) \(\approx\) \(1.48728 - 3.15548i\)
\(L(3)\) 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 \( 1 + (93.5 + 140. i)T \)
good2 \( 1 + (-4.32 + 4.32i)T - 16iT^{2} \)
5 \( 1 + (-26.0 + 26.0i)T - 625iT^{2} \)
7 \( 1 + (16.9 + 16.9i)T + 2.40e3iT^{2} \)
11 \( 1 + (-69.5 - 69.5i)T + 1.46e4iT^{2} \)
17 \( 1 + 166. iT - 8.35e4T^{2} \)
19 \( 1 + (468. - 468. i)T - 1.30e5iT^{2} \)
23 \( 1 + 388. iT - 2.79e5T^{2} \)
29 \( 1 - 767.T + 7.07e5T^{2} \)
31 \( 1 + (66.8 - 66.8i)T - 9.23e5iT^{2} \)
37 \( 1 + (-1.53e3 - 1.53e3i)T + 1.87e6iT^{2} \)
41 \( 1 + (802. - 802. i)T - 2.82e6iT^{2} \)
43 \( 1 - 2.79e3iT - 3.41e6T^{2} \)
47 \( 1 + (33.8 + 33.8i)T + 4.87e6iT^{2} \)
53 \( 1 - 3.13e3T + 7.89e6T^{2} \)
59 \( 1 + (-3.95e3 - 3.95e3i)T + 1.21e7iT^{2} \)
61 \( 1 - 504.T + 1.38e7T^{2} \)
67 \( 1 + (-4.21e3 + 4.21e3i)T - 2.01e7iT^{2} \)
71 \( 1 + (-3.02e3 + 3.02e3i)T - 2.54e7iT^{2} \)
73 \( 1 + (-3.08e3 - 3.08e3i)T + 2.83e7iT^{2} \)
79 \( 1 + 9.37e3T + 3.89e7T^{2} \)
83 \( 1 + (6.22e3 - 6.22e3i)T - 4.74e7iT^{2} \)
89 \( 1 + (8.55e3 + 8.55e3i)T + 6.27e7iT^{2} \)
97 \( 1 + (-2.88e3 + 2.88e3i)T - 8.85e7iT^{2} \)
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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

−12.71799389481583695096736861194, −11.78609262828262982328944831120, −10.27503782985290388095557675540, −9.805405807109031735934208615105, −8.296933532227874282098248556229, −6.34407834587335230421779988860, −5.15001456249282220180933010236, −4.20688789343782670323032484330, −2.52803935147116536265641460351, −1.19570359007844755347932927938, 2.44945120962235741826858481936, 4.01174717875818863068712213828, 5.54372083263006884103328195407, 6.46405493278802585183961410309, 7.02035673387324095913240696747, 8.782084496274092311157639697681, 10.01005748863846006478792114374, 11.23021017894218474015977451399, 12.61544276782272288047155799397, 13.56525072149828387349366973946

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