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Properties

Label 2-117-117.103-c3-0-18
Degree $2$
Conductor $117$
Sign $0.613 - 0.789i$
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  + (2.63 − 1.52i)2-s + (0.490 + 5.17i)3-s + (0.634 − 1.09i)4-s + (4.80 + 2.77i)5-s + (9.16 + 12.8i)6-s + (8.73 − 5.04i)7-s + 20.4i·8-s + (−26.5 + 5.07i)9-s + 16.8·10-s + (−13.9 + 8.04i)11-s + (5.99 + 2.74i)12-s + (24.3 + 40.0i)13-s + (15.3 − 26.6i)14-s + (−11.9 + 26.2i)15-s + (36.2 + 62.8i)16-s + 68.7·17-s + ⋯
L(s)  = 1  + (0.932 − 0.538i)2-s + (0.0944 + 0.995i)3-s + (0.0793 − 0.137i)4-s + (0.429 + 0.248i)5-s + (0.623 + 0.877i)6-s + (0.471 − 0.272i)7-s + 0.905i·8-s + (−0.982 + 0.187i)9-s + 0.534·10-s + (−0.382 + 0.220i)11-s + (0.144 + 0.0660i)12-s + (0.520 + 0.854i)13-s + (0.293 − 0.507i)14-s + (−0.206 + 0.451i)15-s + (0.566 + 0.981i)16-s + 0.980·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.613 - 0.789i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.613 - 0.789i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (103, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.613 - 0.789i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.40919 + 1.17961i\)
\(L(\frac12)\) \(\approx\) \(2.40919 + 1.17961i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.490 - 5.17i)T \)
13 \( 1 + (-24.3 - 40.0i)T \)
good2 \( 1 + (-2.63 + 1.52i)T + (4 - 6.92i)T^{2} \)
5 \( 1 + (-4.80 - 2.77i)T + (62.5 + 108. i)T^{2} \)
7 \( 1 + (-8.73 + 5.04i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (13.9 - 8.04i)T + (665.5 - 1.15e3i)T^{2} \)
17 \( 1 - 68.7T + 4.91e3T^{2} \)
19 \( 1 + 7.26iT - 6.85e3T^{2} \)
23 \( 1 + (-59.2 + 102. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (69.2 + 119. i)T + (-1.21e4 + 2.11e4i)T^{2} \)
31 \( 1 + (-5.73 - 3.31i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + 228. iT - 5.06e4T^{2} \)
41 \( 1 + (-213. - 123. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (48.2 + 83.5i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + (-245. + 141. i)T + (5.19e4 - 8.99e4i)T^{2} \)
53 \( 1 + 214.T + 1.48e5T^{2} \)
59 \( 1 + (-356. - 205. i)T + (1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (156. + 271. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-164. - 95.1i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 - 671. iT - 3.57e5T^{2} \)
73 \( 1 + 734. iT - 3.89e5T^{2} \)
79 \( 1 + (623. + 1.07e3i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 + (-1.14e3 + 661. i)T + (2.85e5 - 4.95e5i)T^{2} \)
89 \( 1 - 720. iT - 7.04e5T^{2} \)
97 \( 1 + (1.41e3 - 819. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.36352209807350165606862244688, −12.11014155373552936839726325760, −11.15383522372458646208615570712, −10.35630426116116002688185397848, −9.112751334133942742235314197638, −7.914341026713716402389474397953, −5.95969056070506553685508906691, −4.78911062193087749824276909740, −3.83348608573293723166796957547, −2.41991335264427025268171437139, 1.23871210275103875416614441948, 3.28420097648975699899143878624, 5.31709780614527427171984989399, 5.81247440969881403396365056720, 7.20669498603781476851717625622, 8.248593594505221937994760229892, 9.604486415577299114849713108823, 11.09171978301255655287694381861, 12.36904022902408516129983755711, 13.12127343861500136454343704764

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