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Properties

Label 2-117-9.2-c2-0-14
Degree $2$
Conductor $117$
Sign $0.996 - 0.0848i$
Analytic cond. $3.18801$
Root an. cond. $1.78550$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.658 + 0.379i)2-s + (1.82 + 2.38i)3-s + (−1.71 − 2.96i)4-s + (7.70 − 4.45i)5-s + (0.294 + 2.26i)6-s + (−0.0636 + 0.110i)7-s − 5.64i·8-s + (−2.34 + 8.68i)9-s + 6.76·10-s + (−4.42 − 2.55i)11-s + (3.94 − 9.48i)12-s + (1.80 + 3.12i)13-s + (−0.0837 + 0.0483i)14-s + (24.6 + 10.2i)15-s + (−4.70 + 8.14i)16-s + 23.2i·17-s + ⋯
L(s)  = 1  + (0.329 + 0.189i)2-s + (0.607 + 0.794i)3-s + (−0.427 − 0.740i)4-s + (1.54 − 0.890i)5-s + (0.0491 + 0.376i)6-s + (−0.00909 + 0.0157i)7-s − 0.705i·8-s + (−0.261 + 0.965i)9-s + 0.676·10-s + (−0.402 − 0.232i)11-s + (0.328 − 0.790i)12-s + (0.138 + 0.240i)13-s + (−0.00598 + 0.00345i)14-s + (1.64 + 0.683i)15-s + (−0.293 + 0.508i)16-s + 1.36i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0848i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.996 - 0.0848i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.996 - 0.0848i$
Analytic conductor: \(3.18801\)
Root analytic conductor: \(1.78550\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (92, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :1),\ 0.996 - 0.0848i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.07783 + 0.0882852i\)
\(L(\frac12)\) \(\approx\) \(2.07783 + 0.0882852i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.82 - 2.38i)T \)
13 \( 1 + (-1.80 - 3.12i)T \)
good2 \( 1 + (-0.658 - 0.379i)T + (2 + 3.46i)T^{2} \)
5 \( 1 + (-7.70 + 4.45i)T + (12.5 - 21.6i)T^{2} \)
7 \( 1 + (0.0636 - 0.110i)T + (-24.5 - 42.4i)T^{2} \)
11 \( 1 + (4.42 + 2.55i)T + (60.5 + 104. i)T^{2} \)
17 \( 1 - 23.2iT - 289T^{2} \)
19 \( 1 - 9.50T + 361T^{2} \)
23 \( 1 + (22.0 - 12.7i)T + (264.5 - 458. i)T^{2} \)
29 \( 1 + (20.0 + 11.5i)T + (420.5 + 728. i)T^{2} \)
31 \( 1 + (15.4 + 26.8i)T + (-480.5 + 832. i)T^{2} \)
37 \( 1 - 7.35T + 1.36e3T^{2} \)
41 \( 1 + (-32.4 + 18.7i)T + (840.5 - 1.45e3i)T^{2} \)
43 \( 1 + (8.05 - 13.9i)T + (-924.5 - 1.60e3i)T^{2} \)
47 \( 1 + (66.9 + 38.6i)T + (1.10e3 + 1.91e3i)T^{2} \)
53 \( 1 - 49.0iT - 2.80e3T^{2} \)
59 \( 1 + (40.3 - 23.3i)T + (1.74e3 - 3.01e3i)T^{2} \)
61 \( 1 + (-57.3 + 99.3i)T + (-1.86e3 - 3.22e3i)T^{2} \)
67 \( 1 + (-66.4 - 115. i)T + (-2.24e3 + 3.88e3i)T^{2} \)
71 \( 1 + 72.0iT - 5.04e3T^{2} \)
73 \( 1 - 18.8T + 5.32e3T^{2} \)
79 \( 1 + (0.111 - 0.193i)T + (-3.12e3 - 5.40e3i)T^{2} \)
83 \( 1 + (-50.8 - 29.3i)T + (3.44e3 + 5.96e3i)T^{2} \)
89 \( 1 - 100. iT - 7.92e3T^{2} \)
97 \( 1 + (-4.44 + 7.69i)T + (-4.70e3 - 8.14e3i)T^{2} \)
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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

−13.53133237637794225438307547172, −12.82795874499634628804751470104, −10.82688769066998586372393699730, −9.809004070722368301296865940357, −9.372297521455157161238248073259, −8.241865770152329023661088847810, −5.99754096297738404441083600434, −5.36357211210748377459527720640, −4.09552885431623171307839907861, −1.85799524516376930070861282219, 2.23854030147246654969040221107, 3.22233761427317094423817511078, 5.34070827353504039604506403099, 6.70679211777142930532791318104, 7.72066637654356085899349741018, 9.057898875487896390199161685354, 9.942828283384208090393479181858, 11.39479559524943314738933105058, 12.62736198962073009417626886881, 13.37598291331939246532023474547

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