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Properties

Label 2-336-7.3-c6-0-42
Degree $2$
Conductor $336$
Sign $0.763 + 0.645i$
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 + (180. − 104. i)5-s + (327. + 102. i)7-s + (121.5 + 210. i)9-s + (1.03e3 − 1.79e3i)11-s − 156. i·13-s + 3.25e3·15-s + (634. + 366. i)17-s + (6.37e3 − 3.68e3i)19-s + (3.61e3 + 3.93e3i)21-s + (−3.23e3 − 5.59e3i)23-s + (1.39e4 − 2.42e4i)25-s + 3.78e3i·27-s − 50.2·29-s + (−590. − 340. i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (1.44 − 0.835i)5-s + (0.954 + 0.299i)7-s + (0.166 + 0.288i)9-s + (0.779 − 1.35i)11-s − 0.0713i·13-s + 0.964·15-s + (0.129 + 0.0745i)17-s + (0.929 − 0.536i)19-s + (0.390 + 0.425i)21-s + (−0.265 − 0.459i)23-s + (0.895 − 1.55i)25-s + 0.192i·27-s − 0.00206·29-s + (−0.0198 − 0.0114i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.763 + 0.645i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.763 + 0.645i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ 0.763 + 0.645i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(4.363313475\)
\(L(\frac12)\) \(\approx\) \(4.363313475\)
\(L(4)\) 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 \( 1 + (-327. - 102. i)T \)
good5 \( 1 + (-180. + 104. i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-1.03e3 + 1.79e3i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 156. iT - 4.82e6T^{2} \)
17 \( 1 + (-634. - 366. i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-6.37e3 + 3.68e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (3.23e3 + 5.59e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 50.2T + 5.94e8T^{2} \)
31 \( 1 + (590. + 340. i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (2.56e4 + 4.43e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 1.14e5iT - 4.75e9T^{2} \)
43 \( 1 + 1.41e5T + 6.32e9T^{2} \)
47 \( 1 + (-1.23e5 + 7.14e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (9.36e4 - 1.62e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (3.33e5 + 1.92e5i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (2.70e5 - 1.55e5i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-5.74e4 + 9.95e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 1.45e5T + 1.28e11T^{2} \)
73 \( 1 + (4.38e4 + 2.53e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (1.39e5 + 2.41e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 7.18e5iT - 3.26e11T^{2} \)
89 \( 1 + (-7.74e4 + 4.47e4i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.09e6iT - 8.32e11T^{2} \)
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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

−10.29574642929145299646821247120, −9.216616068757283421148579618209, −8.847132473594232655409312926332, −7.889892876644588847545194162460, −6.29982315076966289018319160324, −5.46342587340656256941057032212, −4.60698640780199423675067676873, −3.11007791849028956064240197728, −1.83543051376770231004041148350, −0.979049675783909487764060367416, 1.51817478705856091658722785584, 1.93685944961274035503779450523, 3.30907767618961575649098180370, 4.71685343029514655191751849587, 5.86829096727426167055697324486, 6.94509933110237676163978696561, 7.57405845336809529840290242384, 8.921666406274123670594908653085, 9.813414005315226495096239778665, 10.35107023508749367132529955795

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