Properties

Label 2-336-7.3-c6-0-37
Degree $2$
Conductor $336$
Sign $0.494 + 0.869i$
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 + (151. − 87.3i)5-s + (271. + 209. i)7-s + (121.5 + 210. i)9-s + (−92.6 + 160. i)11-s − 3.98e3i·13-s − 2.72e3·15-s + (6.10e3 + 3.52e3i)17-s + (4.06e3 − 2.34e3i)19-s + (−2.03e3 − 4.94e3i)21-s + (−3.32e3 − 5.75e3i)23-s + (7.45e3 − 1.29e4i)25-s − 3.78e3i·27-s + 1.93e4·29-s + (−2.07e4 − 1.19e4i)31-s + ⋯
L(s)  = 1  + (−0.5 − 0.288i)3-s + (1.21 − 0.699i)5-s + (0.791 + 0.610i)7-s + (0.166 + 0.288i)9-s + (−0.0696 + 0.120i)11-s − 1.81i·13-s − 0.807·15-s + (1.24 + 0.716i)17-s + (0.592 − 0.342i)19-s + (−0.219 − 0.533i)21-s + (−0.273 − 0.472i)23-s + (0.477 − 0.826i)25-s − 0.192i·27-s + 0.794·29-s + (−0.696 − 0.402i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.494 + 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.494 + 0.869i$
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: \((2,\ 336,\ (\ :3),\ 0.494 + 0.869i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.769181906\)
\(L(\frac12)\) \(\approx\) \(2.769181906\)
\(L(4)\) 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 \( 1 + (-271. - 209. i)T \)
good5 \( 1 + (-151. + 87.3i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (92.6 - 160. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 + 3.98e3iT - 4.82e6T^{2} \)
17 \( 1 + (-6.10e3 - 3.52e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (-4.06e3 + 2.34e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (3.32e3 + 5.75e3i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 - 1.93e4T + 5.94e8T^{2} \)
31 \( 1 + (2.07e4 + 1.19e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-1.97e4 - 3.42e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 4.62e4iT - 4.75e9T^{2} \)
43 \( 1 - 9.31e4T + 6.32e9T^{2} \)
47 \( 1 + (1.29e5 - 7.47e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-5.09e4 + 8.81e4i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (1.31e5 + 7.61e4i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (1.45e5 - 8.42e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-3.05e4 + 5.29e4i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 4.85e5T + 1.28e11T^{2} \)
73 \( 1 + (7.73e3 + 4.46e3i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (-2.21e5 - 3.84e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 + 5.59e5iT - 3.26e11T^{2} \)
89 \( 1 + (1.79e5 - 1.03e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 + 1.23e6iT - 8.32e11T^{2} \)
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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

−10.29807404904639636003384840934, −9.600153738049999452860201665358, −8.370913962867096883445091918849, −7.73541051552201040880301103111, −6.09050110198714279636898546172, −5.54626217360150459662704474493, −4.82098147948863119695084896522, −2.91231177432675195020297124659, −1.65306985887703798532512550833, −0.792401737839959956913222812000, 1.12829078372361197744093548398, 2.13836818826775140770545242736, 3.65032224183720518685858852109, 4.88240576972229196938271605166, 5.79345561090440125769604594329, 6.78407272019382790997046934611, 7.62241078256447443628426388559, 9.160233040032686194459541999359, 9.844250994824283837797684854098, 10.65439749239403354750108714364

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