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Properties

Label 2-336-7.5-c6-0-9
Degree $2$
Conductor $336$
Sign $0.629 - 0.776i$
Analytic cond. $77.2981$
Root an. cond. $8.79193$
Motivic weight $6$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (13.5 − 7.79i)3-s + (−118. − 68.4i)5-s + (302. − 162. i)7-s + (121.5 − 210. i)9-s + (−847. − 1.46e3i)11-s + 3.57e3i·13-s − 2.13e3·15-s + (−6.68e3 + 3.86e3i)17-s + (5.56e3 + 3.21e3i)19-s + (2.81e3 − 4.54e3i)21-s + (−9.56e3 + 1.65e4i)23-s + (1.56e3 + 2.70e3i)25-s − 3.78e3i·27-s − 2.98e3·29-s + (1.72e4 − 9.97e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−0.948 − 0.547i)5-s + (0.880 − 0.473i)7-s + (0.166 − 0.288i)9-s + (−0.636 − 1.10i)11-s + 1.62i·13-s − 0.632·15-s + (−1.36 + 0.785i)17-s + (0.811 + 0.468i)19-s + (0.303 − 0.490i)21-s + (−0.785 + 1.36i)23-s + (0.100 + 0.173i)25-s − 0.192i·27-s − 0.122·29-s + (0.579 − 0.334i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.629 - 0.776i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ 0.629 - 0.776i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.450327981\)
\(L(\frac12)\) \(\approx\) \(1.450327981\)
\(L(4)\) 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 \( 1 + (-13.5 + 7.79i)T \)
7 \( 1 + (-302. + 162. i)T \)
good5 \( 1 + (118. + 68.4i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (847. + 1.46e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 3.57e3iT - 4.82e6T^{2} \)
17 \( 1 + (6.68e3 - 3.86e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.56e3 - 3.21e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (9.56e3 - 1.65e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 2.98e3T + 5.94e8T^{2} \)
31 \( 1 + (-1.72e4 + 9.97e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-2.98e4 + 5.17e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 3.35e3iT - 4.75e9T^{2} \)
43 \( 1 - 2.60e4T + 6.32e9T^{2} \)
47 \( 1 + (2.07e4 + 1.19e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-8.46e4 - 1.46e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (2.63e5 - 1.52e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-1.04e5 - 6.05e4i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-2.56e5 - 4.43e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 - 5.55e5T + 1.28e11T^{2} \)
73 \( 1 + (-1.24e5 + 7.17e4i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (8.64e4 - 1.49e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 5.02e5iT - 3.26e11T^{2} \)
89 \( 1 + (4.35e4 + 2.51e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 - 6.41e5iT - 8.32e11T^{2} \)
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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

−10.98647675071115102156586483940, −9.523855949128592519829879605610, −8.527955355192376698563611465916, −7.995623137908714398569257504149, −7.12130236319785212677385551583, −5.77729660163796866660515307321, −4.36508185125665187275494219394, −3.81302047030187173490520838270, −2.14162766609032694377845098278, −1.00690902128132440733403977161, 0.35775701372814124354656329601, 2.24556532937401004538529715955, 3.05589216078403165141918129663, 4.46558907667666042671494214843, 5.14708539179144050589357583402, 6.79454282277325269000008278504, 7.84356374401103344550198913963, 8.210624645445464108839146904006, 9.495383357168764803629114625955, 10.49782173141085104320501617489

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