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Properties

Label 2-336-7.6-c8-0-34
Degree $2$
Conductor $336$
Sign $-0.220 - 0.975i$
Analytic cond. $136.879$
Root an. cond. $11.6995$
Motivic weight $8$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 46.7i·3-s + 1.18e3i·5-s + (2.34e3 − 530. i)7-s − 2.18e3·9-s + 1.63e4·11-s + 3.10e4i·13-s + 5.53e4·15-s + 8.79e4i·17-s + 2.14e5i·19-s + (−2.48e4 − 1.09e5i)21-s + 4.08e5·23-s − 1.01e6·25-s + 1.02e5i·27-s + 6.16e5·29-s − 8.51e5i·31-s + ⋯
L(s)  = 1  − 0.577i·3-s + 1.89i·5-s + (0.975 − 0.220i)7-s − 0.333·9-s + 1.11·11-s + 1.08i·13-s + 1.09·15-s + 1.05i·17-s + 1.64i·19-s + (−0.127 − 0.563i)21-s + 1.46·23-s − 2.59·25-s + 0.192i·27-s + 0.871·29-s − 0.922i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.220 - 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.220 - 0.975i$
Analytic conductor: \(136.879\)
Root analytic conductor: \(11.6995\)
Motivic weight: \(8\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :4),\ -0.220 - 0.975i)\)

Particular Values

\(L(\frac{9}{2})\) \(\approx\) \(2.899905354\)
\(L(\frac12)\) \(\approx\) \(2.899905354\)
\(L(5)\) 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 \( 1 + 46.7iT \)
7 \( 1 + (-2.34e3 + 530. i)T \)
good5 \( 1 - 1.18e3iT - 3.90e5T^{2} \)
11 \( 1 - 1.63e4T + 2.14e8T^{2} \)
13 \( 1 - 3.10e4iT - 8.15e8T^{2} \)
17 \( 1 - 8.79e4iT - 6.97e9T^{2} \)
19 \( 1 - 2.14e5iT - 1.69e10T^{2} \)
23 \( 1 - 4.08e5T + 7.83e10T^{2} \)
29 \( 1 - 6.16e5T + 5.00e11T^{2} \)
31 \( 1 + 8.51e5iT - 8.52e11T^{2} \)
37 \( 1 - 2.85e6T + 3.51e12T^{2} \)
41 \( 1 - 4.79e6iT - 7.98e12T^{2} \)
43 \( 1 + 1.35e6T + 1.16e13T^{2} \)
47 \( 1 + 4.66e6iT - 2.38e13T^{2} \)
53 \( 1 + 3.62e5T + 6.22e13T^{2} \)
59 \( 1 + 1.41e7iT - 1.46e14T^{2} \)
61 \( 1 - 2.57e6iT - 1.91e14T^{2} \)
67 \( 1 + 1.48e6T + 4.06e14T^{2} \)
71 \( 1 - 1.02e7T + 6.45e14T^{2} \)
73 \( 1 + 3.78e7iT - 8.06e14T^{2} \)
79 \( 1 + 6.82e7T + 1.51e15T^{2} \)
83 \( 1 + 1.48e7iT - 2.25e15T^{2} \)
89 \( 1 + 7.13e7iT - 3.93e15T^{2} \)
97 \( 1 - 6.81e7iT - 7.83e15T^{2} \)
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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

−10.61782926222306948720488440527, −9.657362320474219990087350625785, −8.317035157307285929150267855469, −7.49551141675388988693264906070, −6.59197672692256084515351370967, −6.08863437404716483231700414190, −4.30967626362660777646858975421, −3.36596631205814088934816709883, −2.10255447541348341572094379838, −1.31730479705441760689026592894, 0.66014885023801774867592173076, 1.16659474533479252824065392073, 2.76836635965299390457523875192, 4.34513256311549074411772632800, 4.89591754737802704283219031295, 5.57112530062455874936647636461, 7.19697415835560828662335190549, 8.444969208065267409423095075863, 8.916661490290117945764198209205, 9.584515306603310586023708055223

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