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Properties

Label 2-117-13.9-c3-0-6
Degree $2$
Conductor $117$
Sign $0.686 - 0.727i$
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  + (−0.219 + 0.379i)2-s + (3.90 + 6.76i)4-s + 17.8·5-s + (−2.71 − 4.70i)7-s − 6.93·8-s + (−3.90 + 6.76i)10-s + (−11.2 + 19.4i)11-s + (21.9 − 41.4i)13-s + 2.38·14-s + (−29.7 + 51.4i)16-s + (33.9 + 58.8i)17-s + (40.4 + 69.9i)19-s + (69.5 + 120. i)20-s + (−4.91 − 8.51i)22-s + (70.2 − 121. i)23-s + ⋯
L(s)  = 1  + (−0.0775 + 0.134i)2-s + (0.487 + 0.845i)4-s + 1.59·5-s + (−0.146 − 0.254i)7-s − 0.306·8-s + (−0.123 + 0.213i)10-s + (−0.307 + 0.532i)11-s + (0.468 − 0.883i)13-s + 0.0455·14-s + (−0.464 + 0.804i)16-s + (0.484 + 0.839i)17-s + (0.487 + 0.844i)19-s + (0.777 + 1.34i)20-s + (−0.0476 − 0.0825i)22-s + (0.637 − 1.10i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 117 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(117\)    =    \(3^{2} \cdot 13\)
Sign: $0.686 - 0.727i$
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: \((2,\ 117,\ (\ :3/2),\ 0.686 - 0.727i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.94176 + 0.837804i\)
\(L(\frac12)\) \(\approx\) \(1.94176 + 0.837804i\)
\(L(\frac{5}{2})\) 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 \( 1 + (-21.9 + 41.4i)T \)
good2 \( 1 + (0.219 - 0.379i)T + (-4 - 6.92i)T^{2} \)
5 \( 1 - 17.8T + 125T^{2} \)
7 \( 1 + (2.71 + 4.70i)T + (-171.5 + 297. i)T^{2} \)
11 \( 1 + (11.2 - 19.4i)T + (-665.5 - 1.15e3i)T^{2} \)
17 \( 1 + (-33.9 - 58.8i)T + (-2.45e3 + 4.25e3i)T^{2} \)
19 \( 1 + (-40.4 - 69.9i)T + (-3.42e3 + 5.94e3i)T^{2} \)
23 \( 1 + (-70.2 + 121. i)T + (-6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + (53.3 - 92.3i)T + (-1.21e4 - 2.11e4i)T^{2} \)
31 \( 1 + 276.T + 2.97e4T^{2} \)
37 \( 1 + (-2.14 + 3.71i)T + (-2.53e4 - 4.38e4i)T^{2} \)
41 \( 1 + (-113. + 197. i)T + (-3.44e4 - 5.96e4i)T^{2} \)
43 \( 1 + (13.7 + 23.8i)T + (-3.97e4 + 6.88e4i)T^{2} \)
47 \( 1 + 318.T + 1.03e5T^{2} \)
53 \( 1 - 67.6T + 1.48e5T^{2} \)
59 \( 1 + (145. + 252. i)T + (-1.02e5 + 1.77e5i)T^{2} \)
61 \( 1 + (331. + 574. i)T + (-1.13e5 + 1.96e5i)T^{2} \)
67 \( 1 + (-212. + 368. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 + (76.4 + 132. i)T + (-1.78e5 + 3.09e5i)T^{2} \)
73 \( 1 - 117.T + 3.89e5T^{2} \)
79 \( 1 - 202.T + 4.93e5T^{2} \)
83 \( 1 + 336.T + 5.71e5T^{2} \)
89 \( 1 + (-359. + 621. i)T + (-3.52e5 - 6.10e5i)T^{2} \)
97 \( 1 + (379. + 657. i)T + (-4.56e5 + 7.90e5i)T^{2} \)
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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

−12.88156688527152923960423572049, −12.63174230772057874677622650216, −10.87598489753538788262207530402, −10.11018681641392328735097563138, −8.899250308668896169275338344025, −7.68955897494403767289870337803, −6.47008286687110832070509249488, −5.43402047012647233619960923298, −3.39527612913671400414698667199, −1.89988549567867406498740456971, 1.39032076750939558753739642488, 2.73499571020779512922929687407, 5.26895554366352358300237449735, 5.97408214764980985726288107685, 7.11475189356734420594907036683, 9.226491902384913317577014782365, 9.547859043958492129996199993371, 10.80909266186872593687105315291, 11.61689724463569074365864273234, 13.24453823317562996741551307785

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