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Properties

Label 2-336-7.5-c6-0-44
Degree $2$
Conductor $336$
Sign $-0.646 - 0.763i$
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 + (−165. − 95.4i)5-s + (103. + 327. i)7-s + (121.5 − 210. i)9-s + (−1.02e3 − 1.77e3i)11-s − 3.05e3i·13-s − 2.97e3·15-s + (2.46e3 − 1.42e3i)17-s + (3.42e3 + 1.97e3i)19-s + (3.94e3 + 3.61e3i)21-s + (330. − 572. i)23-s + (1.04e4 + 1.80e4i)25-s − 3.78e3i·27-s − 9.28e3·29-s + (−2.42e3 + 1.40e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−1.32 − 0.763i)5-s + (0.301 + 0.953i)7-s + (0.166 − 0.288i)9-s + (−0.772 − 1.33i)11-s − 1.39i·13-s − 0.881·15-s + (0.502 − 0.290i)17-s + (0.498 + 0.287i)19-s + (0.425 + 0.389i)21-s + (0.0271 − 0.0470i)23-s + (0.665 + 1.15i)25-s − 0.192i·27-s − 0.380·29-s + (−0.0814 + 0.0470i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.646 - 0.763i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.646 - 0.763i$
Analytic conductor: \(77.2981\)
Root analytic conductor: \(8.79193\)
Motivic weight: \(6\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3),\ -0.646 - 0.763i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(0.2449167605\)
\(L(\frac12)\) \(\approx\) \(0.2449167605\)
\(L(4)\) 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 \( 1 + (-103. - 327. i)T \)
good5 \( 1 + (165. + 95.4i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (1.02e3 + 1.77e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 3.05e3iT - 4.82e6T^{2} \)
17 \( 1 + (-2.46e3 + 1.42e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-3.42e3 - 1.97e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-330. + 572. i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 9.28e3T + 5.94e8T^{2} \)
31 \( 1 + (2.42e3 - 1.40e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-1.84e4 + 3.19e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 6.79e4iT - 4.75e9T^{2} \)
43 \( 1 + 1.23e4T + 6.32e9T^{2} \)
47 \( 1 + (1.27e5 + 7.38e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (1.09e5 + 1.90e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (1.66e5 - 9.62e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-2.88e5 - 1.66e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.74e5 - 3.02e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 3.05e5T + 1.28e11T^{2} \)
73 \( 1 + (2.04e5 - 1.18e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (2.93e5 - 5.07e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 1.06e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.13e4 - 6.52e3i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 2.05e5iT - 8.32e11T^{2} \)
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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

−9.815573988545398635370546464395, −8.491719661265257196285298862009, −8.271142062257484957362413673796, −7.49918352313908966968510523416, −5.78775634342013577594033122929, −5.05724713047983999896346301732, −3.56747416773693534123931692511, −2.80241352832361856579629131376, −1.04794353479130255987115922487, −0.06025961772473725317310648839, 1.73031906484673960688571478489, 3.13410891088126169630686991528, 4.11539058691071740518530753995, 4.77467883345724378373999601725, 6.75747341986568816848895879035, 7.47521173016128201059779163960, 7.980785866176185961403526790597, 9.374732312660174681767179116546, 10.25978270865691124652550174701, 11.08416850454656885667701625282

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