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Properties

Label 2-312-13.9-c5-0-11
Degree $2$
Conductor $312$
Sign $0.996 - 0.0886i$
Analytic cond. $50.0397$
Root an. cond. $7.07387$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (4.5 − 7.79i)3-s − 47.9·5-s + (5.03 + 8.71i)7-s + (−40.5 − 70.1i)9-s + (−90.6 + 156. i)11-s + (−469. − 387. i)13-s + (−215. + 373. i)15-s + (1.17e3 + 2.03e3i)17-s + (−690. − 1.19e3i)19-s + 90.5·21-s + (787. − 1.36e3i)23-s − 825.·25-s − 729·27-s + (−979. + 1.69e3i)29-s + 7.06e3·31-s + ⋯
L(s)  = 1  + (0.288 − 0.499i)3-s − 0.857·5-s + (0.0388 + 0.0672i)7-s + (−0.166 − 0.288i)9-s + (−0.225 + 0.391i)11-s + (−0.771 − 0.636i)13-s + (−0.247 + 0.428i)15-s + (0.986 + 1.70i)17-s + (−0.438 − 0.760i)19-s + 0.0448·21-s + (0.310 − 0.537i)23-s − 0.264·25-s − 0.192·27-s + (−0.216 + 0.374i)29-s + 1.31·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.996 - 0.0886i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 312 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.996 - 0.0886i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(312\)    =    \(2^{3} \cdot 3 \cdot 13\)
Sign: $0.996 - 0.0886i$
Analytic conductor: \(50.0397\)
Root analytic conductor: \(7.07387\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{312} (217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 312,\ (\ :5/2),\ 0.996 - 0.0886i)\)

Particular Values

\(L(3)\) \(\approx\) \(1.582979365\)
\(L(\frac12)\) \(\approx\) \(1.582979365\)
\(L(\frac{7}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-4.5 + 7.79i)T \)
13 \( 1 + (469. + 387. i)T \)
good5 \( 1 + 47.9T + 3.12e3T^{2} \)
7 \( 1 + (-5.03 - 8.71i)T + (-8.40e3 + 1.45e4i)T^{2} \)
11 \( 1 + (90.6 - 156. i)T + (-8.05e4 - 1.39e5i)T^{2} \)
17 \( 1 + (-1.17e3 - 2.03e3i)T + (-7.09e5 + 1.22e6i)T^{2} \)
19 \( 1 + (690. + 1.19e3i)T + (-1.23e6 + 2.14e6i)T^{2} \)
23 \( 1 + (-787. + 1.36e3i)T + (-3.21e6 - 5.57e6i)T^{2} \)
29 \( 1 + (979. - 1.69e3i)T + (-1.02e7 - 1.77e7i)T^{2} \)
31 \( 1 - 7.06e3T + 2.86e7T^{2} \)
37 \( 1 + (7.42e3 - 1.28e4i)T + (-3.46e7 - 6.00e7i)T^{2} \)
41 \( 1 + (-9.01e3 + 1.56e4i)T + (-5.79e7 - 1.00e8i)T^{2} \)
43 \( 1 + (-9.34e3 - 1.61e4i)T + (-7.35e7 + 1.27e8i)T^{2} \)
47 \( 1 - 2.24e4T + 2.29e8T^{2} \)
53 \( 1 + 1.65e3T + 4.18e8T^{2} \)
59 \( 1 + (1.06e4 + 1.83e4i)T + (-3.57e8 + 6.19e8i)T^{2} \)
61 \( 1 + (-1.18e4 - 2.05e4i)T + (-4.22e8 + 7.31e8i)T^{2} \)
67 \( 1 + (1.99e3 - 3.45e3i)T + (-6.75e8 - 1.16e9i)T^{2} \)
71 \( 1 + (-255. - 443. i)T + (-9.02e8 + 1.56e9i)T^{2} \)
73 \( 1 - 4.40e4T + 2.07e9T^{2} \)
79 \( 1 + 4.34e4T + 3.07e9T^{2} \)
83 \( 1 - 9.27e4T + 3.93e9T^{2} \)
89 \( 1 + (-5.06e4 + 8.77e4i)T + (-2.79e9 - 4.83e9i)T^{2} \)
97 \( 1 + (-7.70e4 - 1.33e5i)T + (-4.29e9 + 7.43e9i)T^{2} \)
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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

−10.81134056214536070271199908042, −10.02865717875635270887585388882, −8.684736067154238963282982748665, −7.958008615006207175722682510669, −7.20750610425805003466536450874, −6.02661422148176552939636996482, −4.71756391597257542925032402282, −3.54018203865266282129199304964, −2.33018251756395539242563439539, −0.797744319699961463241124490688, 0.58576978970777126152662309811, 2.48839495288858035313496001491, 3.64874954927995205133373053206, 4.61761537157483905322954809952, 5.72053615347670042434958071153, 7.29448362414430680282520231742, 7.83668837310345823203649605857, 9.043500260982270720999992663075, 9.805236114250800409124262491054, 10.84587599171577309739312446777

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