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Properties

Label 2-117-13.4-c3-0-12
Degree $2$
Conductor $117$
Sign $0.964 + 0.265i$
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  + (3 + 1.73i)2-s + (2 + 3.46i)4-s − 13.8i·5-s + (19.5 − 11.2i)7-s − 13.8i·8-s + (23.9 − 41.5i)10-s + (19.5 + 11.2i)11-s + (−13 + 45.0i)13-s + 78·14-s + (39.9 − 69.2i)16-s + (13.5 + 23.3i)17-s + (−76.5 + 44.1i)19-s + (48.0 − 27.7i)20-s + (39 + 67.5i)22-s + (28.5 − 49.3i)23-s + ⋯
L(s)  = 1  + (1.06 + 0.612i)2-s + (0.250 + 0.433i)4-s − 1.23i·5-s + (1.05 − 0.607i)7-s − 0.612i·8-s + (0.758 − 1.31i)10-s + (0.534 + 0.308i)11-s + (−0.277 + 0.960i)13-s + 1.48·14-s + (0.624 − 1.08i)16-s + (0.192 + 0.333i)17-s + (−0.923 + 0.533i)19-s + (0.536 − 0.309i)20-s + (0.377 + 0.654i)22-s + (0.258 − 0.447i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.964 + 0.265i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.964 + 0.265i$
Analytic conductor: \(6.90322\)
Root analytic conductor: \(2.62739\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{117} (82, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 117,\ (\ :3/2),\ 0.964 + 0.265i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.84194 - 0.383421i\)
\(L(\frac12)\) \(\approx\) \(2.84194 - 0.383421i\)
\(L(\frac{5}{2})\) 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 \( 1 + (13 - 45.0i)T \)
good2 \( 1 + (-3 - 1.73i)T + (4 + 6.92i)T^{2} \)
5 \( 1 + 13.8iT - 125T^{2} \)
7 \( 1 + (-19.5 + 11.2i)T + (171.5 - 297. i)T^{2} \)
11 \( 1 + (-19.5 - 11.2i)T + (665.5 + 1.15e3i)T^{2} \)
17 \( 1 + (-13.5 - 23.3i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (76.5 - 44.1i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-28.5 + 49.3i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (34.5 - 59.7i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 72.7iT - 2.97e4T^{2} \)
37 \( 1 + (34.5 + 19.9i)T + (2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + (-340.5 - 196. i)T + (3.44e4 + 5.96e4i)T^{2} \)
43 \( 1 + (-42.5 - 73.6i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 - 342. iT - 1.03e5T^{2} \)
53 \( 1 + 426T + 1.48e5T^{2} \)
59 \( 1 + (-16.5 + 9.52i)T + (1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-8.5 - 14.7i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-142.5 - 82.2i)T + (1.50e5 + 2.60e5i)T^{2} \)
71 \( 1 + (505.5 - 291. i)T + (1.78e5 - 3.09e5i)T^{2} \)
73 \( 1 - 1.00e3iT - 3.89e5T^{2} \)
79 \( 1 + 1.24e3T + 4.93e5T^{2} \)
83 \( 1 - 426. iT - 5.71e5T^{2} \)
89 \( 1 + (265.5 + 153. i)T + (3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 + (-1.06e3 + 617. i)T + (4.56e5 - 7.90e5i)T^{2} \)
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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

−13.01221302977528308469550986403, −12.42367806003130677447226061151, −11.21467285262399969721925163157, −9.710015530470312956424307225656, −8.542697695022497590944451970739, −7.31787021205226107606993654545, −6.00985419631430106998444668079, −4.64161844728261179857798511611, −4.26309525114901177612276458403, −1.35351404865054548894972742424, 2.25995207767357435164367621395, 3.37837983410957824319240141260, 4.86016382212429959101308102599, 6.01771565550680087053531216290, 7.54048570399965689406601754084, 8.765046303144688749093042001288, 10.50909562866011766445453706213, 11.22647456739344155084107078590, 11.98625923597066009433965016714, 13.09154426700337427850390128865

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