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Properties

Label 2-336-7.3-c6-0-30
Degree $2$
Conductor $336$
Sign $0.821 - 0.569i$
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 + (201. − 116. i)5-s + (−184. + 289. i)7-s + (121.5 + 210. i)9-s + (460. − 796. i)11-s + 536. i·13-s + 3.62e3·15-s + (2.61e3 + 1.50e3i)17-s + (−7.37e3 + 4.25e3i)19-s + (−4.74e3 + 2.46e3i)21-s + (1.05e4 + 1.82e4i)23-s + (1.92e4 − 3.33e4i)25-s + 3.78e3i·27-s − 3.91e3·29-s + (1.99e4 + 1.15e4i)31-s + ⋯
L(s)  = 1  + (0.5 + 0.288i)3-s + (1.61 − 0.930i)5-s + (−0.537 + 0.843i)7-s + (0.166 + 0.288i)9-s + (0.345 − 0.598i)11-s + 0.244i·13-s + 1.07·15-s + (0.531 + 0.306i)17-s + (−1.07 + 0.620i)19-s + (−0.512 + 0.266i)21-s + (0.865 + 1.49i)23-s + (1.23 − 2.13i)25-s + 0.192i·27-s − 0.160·29-s + (0.669 + 0.386i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.821 - 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.821 - 0.569i$
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: \((2,\ 336,\ (\ :3),\ 0.821 - 0.569i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(3.580846333\)
\(L(\frac12)\) \(\approx\) \(3.580846333\)
\(L(4)\) 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 \( 1 + (184. - 289. i)T \)
good5 \( 1 + (-201. + 116. i)T + (7.81e3 - 1.35e4i)T^{2} \)
11 \( 1 + (-460. + 796. i)T + (-8.85e5 - 1.53e6i)T^{2} \)
13 \( 1 - 536. iT - 4.82e6T^{2} \)
17 \( 1 + (-2.61e3 - 1.50e3i)T + (1.20e7 + 2.09e7i)T^{2} \)
19 \( 1 + (7.37e3 - 4.25e3i)T + (2.35e7 - 4.07e7i)T^{2} \)
23 \( 1 + (-1.05e4 - 1.82e4i)T + (-7.40e7 + 1.28e8i)T^{2} \)
29 \( 1 + 3.91e3T + 5.94e8T^{2} \)
31 \( 1 + (-1.99e4 - 1.15e4i)T + (4.43e8 + 7.68e8i)T^{2} \)
37 \( 1 + (-2.20e4 - 3.82e4i)T + (-1.28e9 + 2.22e9i)T^{2} \)
41 \( 1 - 5.66e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.20e5T + 6.32e9T^{2} \)
47 \( 1 + (1.31e5 - 7.58e4i)T + (5.38e9 - 9.33e9i)T^{2} \)
53 \( 1 + (-1.15e5 + 2.00e5i)T + (-1.10e10 - 1.91e10i)T^{2} \)
59 \( 1 + (-7.46e3 - 4.30e3i)T + (2.10e10 + 3.65e10i)T^{2} \)
61 \( 1 + (-6.85e4 + 3.95e4i)T + (2.57e10 - 4.46e10i)T^{2} \)
67 \( 1 + (-1.56e5 + 2.71e5i)T + (-4.52e10 - 7.83e10i)T^{2} \)
71 \( 1 - 2.12e5T + 1.28e11T^{2} \)
73 \( 1 + (1.98e5 + 1.14e5i)T + (7.56e10 + 1.31e11i)T^{2} \)
79 \( 1 + (3.79e5 + 6.56e5i)T + (-1.21e11 + 2.10e11i)T^{2} \)
83 \( 1 - 1.61e5iT - 3.26e11T^{2} \)
89 \( 1 + (4.86e5 - 2.80e5i)T + (2.48e11 - 4.30e11i)T^{2} \)
97 \( 1 - 1.99e5iT - 8.32e11T^{2} \)
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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

−10.24539084417822191837095138705, −9.534432817149240130561383804477, −8.970929405577895376554801628528, −8.179093160524059879038335979432, −6.41909300808388260102943518129, −5.77771017195591010877036933928, −4.82985716386963032389033655512, −3.33242808122474221995327403571, −2.15424717468343957309587729491, −1.19304548048622912135470727506, 0.826409653357325263678098614459, 2.18996415811791980342764399688, 2.91077994180661801034459583059, 4.32233530381239334121209983467, 5.82781536002357507367819742290, 6.74635232800308954414880436066, 7.22461548499002216142778516109, 8.763647067910604753838945117059, 9.669309351659236015145988274592, 10.29072684864144793737881429859

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