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Properties

Label 2-117-13.9-c3-0-5
Degree $2$
Conductor $117$
Sign $0.998 - 0.0583i$
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  + (1.18 − 2.04i)2-s + (1.21 + 2.09i)4-s − 6.42·5-s + (14.7 + 25.5i)7-s + 24.6·8-s + (−7.58 + 13.1i)10-s + (−0.312 + 0.541i)11-s + (44.3 − 15.0i)13-s + 69.6·14-s + (19.3 − 33.5i)16-s + (43.8 + 75.9i)17-s + (−41.4 − 71.7i)19-s + (−7.77 − 13.4i)20-s + (0.737 + 1.27i)22-s + (−37.3 + 64.7i)23-s + ⋯
L(s)  = 1  + (0.417 − 0.723i)2-s + (0.151 + 0.262i)4-s − 0.574·5-s + (0.796 + 1.37i)7-s + 1.08·8-s + (−0.239 + 0.415i)10-s + (−0.00856 + 0.0148i)11-s + (0.947 − 0.320i)13-s + 1.32·14-s + (0.302 − 0.524i)16-s + (0.625 + 1.08i)17-s + (−0.499 − 0.865i)19-s + (−0.0869 − 0.150i)20-s + (0.00714 + 0.0123i)22-s + (−0.339 + 0.587i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0583i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.998 - 0.0583i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (100, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.998 - 0.0583i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.18602 + 0.0638137i\)
\(L(\frac12)\) \(\approx\) \(2.18602 + 0.0638137i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-44.3 + 15.0i)T \)
good2 \( 1 + (-1.18 + 2.04i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 + 6.42T + 125T^{2} \)
7 \( 1 + (-14.7 - 25.5i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (0.312 - 0.541i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-43.8 - 75.9i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (41.4 + 71.7i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (37.3 - 64.7i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (-113. + 196. i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 - 173.T + 2.97e4T^{2} \)
37 \( 1 + (56.0 - 97.0i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (133. - 231. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (191. + 332. i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 337.T + 1.03e5T^{2} \)
53 \( 1 - 146.T + 1.48e5T^{2} \)
59 \( 1 + (264. + 458. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (101. + 176. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (60.7 - 105. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (330. + 572. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 167.T + 3.89e5T^{2} \)
79 \( 1 + 101.T + 4.93e5T^{2} \)
83 \( 1 - 506.T + 5.71e5T^{2} \)
89 \( 1 + (-701. + 1.21e3i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (951. + 1.64e3i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.83147267710624291003170168721, −11.79766899478712845110486383182, −11.47810883436927587878906573349, −10.22343007804405785282023616305, −8.492559856522706044029343936062, −7.956108371902314224928941425844, −6.14076023493876883959363060021, −4.68219831598339837118607954018, −3.33687588725590834465937429802, −1.87774614263134023708360343707, 1.23255765403806199766730448482, 3.90940696282279618139763199660, 4.93011126874223981467261304727, 6.42715575003250817070862800671, 7.44125155050749456736432862176, 8.274063292998129584602885415027, 10.15532187647425938935001893259, 10.90656728160953247897979912636, 11.93656817173103096162662130226, 13.54001662943221604393588887461

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