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Properties

Label 2-117-13.8-c4-0-7
Degree $2$
Conductor $117$
Sign $0.463 + 0.886i$
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  + (−4.67 − 4.67i)2-s + 27.7i·4-s + (−13.1 − 13.1i)5-s + (−30.2 + 30.2i)7-s + (55.1 − 55.1i)8-s + 122. i·10-s + (−58.8 + 58.8i)11-s + (120. − 118. i)13-s + 282.·14-s − 71.7·16-s + 257. i·17-s + (336. + 336. i)19-s + (365. − 365. i)20-s + 550.·22-s − 486. i·23-s + ⋯
L(s)  = 1  + (−1.16 − 1.16i)2-s + 1.73i·4-s + (−0.525 − 0.525i)5-s + (−0.616 + 0.616i)7-s + (0.862 − 0.862i)8-s + 1.22i·10-s + (−0.485 + 0.485i)11-s + (0.713 − 0.700i)13-s + 1.44·14-s − 0.280·16-s + 0.891i·17-s + (0.931 + 0.931i)19-s + (0.913 − 0.913i)20-s + 1.13·22-s − 0.919i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.463 + 0.886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.463 + 0.886i$
Analytic conductor: \(12.0942\)
Root analytic conductor: \(3.47768\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :2),\ 0.463 + 0.886i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.579703 - 0.351037i\)
\(L(\frac12)\) \(\approx\) \(0.579703 - 0.351037i\)
\(L(3)\) 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 \( 1 + (-120. + 118. i)T \)
good2 \( 1 + (4.67 + 4.67i)T + 16iT^{2} \)
5 \( 1 + (13.1 + 13.1i)T + 625iT^{2} \)
7 \( 1 + (30.2 - 30.2i)T - 2.40e3iT^{2} \)
11 \( 1 + (58.8 - 58.8i)T - 1.46e4iT^{2} \)
17 \( 1 - 257. iT - 8.35e4T^{2} \)
19 \( 1 + (-336. - 336. i)T + 1.30e5iT^{2} \)
23 \( 1 + 486. iT - 2.79e5T^{2} \)
29 \( 1 - 771.T + 7.07e5T^{2} \)
31 \( 1 + (1.30e3 + 1.30e3i)T + 9.23e5iT^{2} \)
37 \( 1 + (-1.05e3 + 1.05e3i)T - 1.87e6iT^{2} \)
41 \( 1 + (-1.12e3 - 1.12e3i)T + 2.82e6iT^{2} \)
43 \( 1 - 1.93e3iT - 3.41e6T^{2} \)
47 \( 1 + (-65.7 + 65.7i)T - 4.87e6iT^{2} \)
53 \( 1 - 5.04e3T + 7.89e6T^{2} \)
59 \( 1 + (-206. + 206. i)T - 1.21e7iT^{2} \)
61 \( 1 - 6.99e3T + 1.38e7T^{2} \)
67 \( 1 + (939. + 939. i)T + 2.01e7iT^{2} \)
71 \( 1 + (-3.04e3 - 3.04e3i)T + 2.54e7iT^{2} \)
73 \( 1 + (840. - 840. i)T - 2.83e7iT^{2} \)
79 \( 1 - 4.04e3T + 3.89e7T^{2} \)
83 \( 1 + (2.06e3 + 2.06e3i)T + 4.74e7iT^{2} \)
89 \( 1 + (6.03e3 - 6.03e3i)T - 6.27e7iT^{2} \)
97 \( 1 + (-5.57e3 - 5.57e3i)T + 8.85e7iT^{2} \)
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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

−12.53449321430725594116040441379, −11.48336382699568594418643173896, −10.42633647300190843544129501570, −9.589847287898001844860813216929, −8.498988484327483505824179778447, −7.80892939805131685040570731229, −5.84535187302915501060309906919, −3.86884906667257488315938462120, −2.47986450412030259638970478020, −0.78726964986329520113861952905, 0.70744340366295474961971927299, 3.41399154116419173133312467458, 5.45559042114160917909822527866, 6.92849611549900048800763536669, 7.27827807914165490498875466978, 8.630091635664709055293353494718, 9.535476291007177419463456162066, 10.62401337253011817607749811088, 11.59318925264846001675450754963, 13.37758448928896960896145643389

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