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Properties

Label 2-336-7.3-c6-0-20
Degree $2$
Conductor $336$
Sign $0.726 - 0.687i$
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 + (41.1 − 23.7i)5-s + (138. − 313. i)7-s + (121.5 + 210. i)9-s + (287. − 497. i)11-s + 2.47e3i·13-s − 741.·15-s + (6.31e3 + 3.64e3i)17-s + (−8.32e3 + 4.80e3i)19-s + (−4.31e3 + 3.15e3i)21-s + (−6.99e3 − 1.21e4i)23-s + (−6.68e3 + 1.15e4i)25-s − 3.78e3i·27-s − 7.40e3·29-s + (4.31e4 + 2.48e4i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (0.329 − 0.190i)5-s + (0.404 − 0.914i)7-s + (0.166 + 0.288i)9-s + (0.215 − 0.373i)11-s + 1.12i·13-s − 0.219·15-s + (1.28 + 0.742i)17-s + (−1.21 + 0.700i)19-s + (−0.466 + 0.340i)21-s + (−0.575 − 0.996i)23-s + (−0.427 + 0.740i)25-s − 0.192i·27-s − 0.303·29-s + (1.44 + 0.835i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.726 - 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.726 - 0.687i$
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: \((2,\ 336,\ (\ :3),\ 0.726 - 0.687i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(1.603550003\)
\(L(\frac12)\) \(\approx\) \(1.603550003\)
\(L(4)\) 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 \( 1 + (-138. + 313. i)T \)
good5 \( 1 + (-41.1 + 23.7i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-287. + 497. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 2.47e3iT - 4.82e6T^{2} \)
17 \( 1 + (-6.31e3 - 3.64e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (8.32e3 - 4.80e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (6.99e3 + 1.21e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 7.40e3T + 5.94e8T^{2} \)
31 \( 1 + (-4.31e4 - 2.48e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (2.02e4 + 3.50e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 3.60e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.61e4T + 6.32e9T^{2} \)
47 \( 1 + (5.79e4 - 3.34e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (4.00e4 - 6.92e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-1.49e5 - 8.60e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-8.99e4 + 5.19e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (8.11e4 - 1.40e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 + 5.22e5T + 1.28e11T^{2} \)
73 \( 1 + (-7.86e4 - 4.54e4i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.35e5 - 4.08e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 3.61e5iT - 3.26e11T^{2} \)
89 \( 1 + (-7.06e5 + 4.07e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.50e6iT - 8.32e11T^{2} \)
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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

−10.58537692239247397876596561455, −9.978298169574311296845451967221, −8.647461001361105594110659381271, −7.80436734806784406814655682563, −6.67353768929879307018635074123, −5.92516681590492726448445137970, −4.64921053788265195105840728814, −3.73473392974416623218931195310, −1.89495188298140719323778105020, −1.02655098520065039107888128647, 0.45346445064374010231728096423, 1.97000724635430372484084828460, 3.15307979587169245163951080025, 4.62990705075489687960873438059, 5.54652307696087716302375046524, 6.29860175353403543558528636978, 7.63187729396524251997540165240, 8.551027383454266945196373044975, 9.745686657025717528964246152703, 10.25944283964784167647278801568

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