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Properties

Label 2-336-7.5-c6-0-22
Degree $2$
Conductor $336$
Sign $0.886 + 0.463i$
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 + (53.9 + 31.1i)5-s + (−218. − 264. i)7-s + (121.5 − 210. i)9-s + (9.71 + 16.8i)11-s + 1.64e3i·13-s + 970.·15-s + (−186. + 107. i)17-s + (8.23e3 + 4.75e3i)19-s + (−5.01e3 − 1.86e3i)21-s + (−7.22e3 + 1.25e4i)23-s + (−5.87e3 − 1.01e4i)25-s − 3.78e3i·27-s + 4.30e4·29-s + (7.79e3 − 4.50e3i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (0.431 + 0.249i)5-s + (−0.637 − 0.770i)7-s + (0.166 − 0.288i)9-s + (0.00730 + 0.0126i)11-s + 0.747i·13-s + 0.287·15-s + (−0.0378 + 0.0218i)17-s + (1.20 + 0.693i)19-s + (−0.541 − 0.200i)21-s + (−0.593 + 1.02i)23-s + (−0.375 − 0.651i)25-s − 0.192i·27-s + 1.76·29-s + (0.261 − 0.151i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.886 + 0.463i$
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: \((2,\ 336,\ (\ :3),\ 0.886 + 0.463i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.692273778\)
\(L(\frac12)\) \(\approx\) \(2.692273778\)
\(L(4)\) 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 \( 1 + (218. + 264. i)T \)
good5 \( 1 + (-53.9 - 31.1i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-9.71 - 16.8i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 - 1.64e3iT - 4.82e6T^{2} \)
17 \( 1 + (186. - 107. i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-8.23e3 - 4.75e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (7.22e3 - 1.25e4i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 - 4.30e4T + 5.94e8T^{2} \)
31 \( 1 + (-7.79e3 + 4.50e3i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (-1.65e4 + 2.86e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 + 7.37e4iT - 4.75e9T^{2} \)
43 \( 1 + 4.76e3T + 6.32e9T^{2} \)
47 \( 1 + (-6.37e4 - 3.68e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (1.19e5 + 2.06e5i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-2.82e5 + 1.63e5i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-3.50e5 - 2.02e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-1.09e5 - 1.90e5i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 3.50e5T + 1.28e11T^{2} \)
73 \( 1 + (1.74e5 - 1.00e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-1.97e5 + 3.42e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 + 1.32e4iT - 3.26e11T^{2} \)
89 \( 1 + (1.99e5 + 1.15e5i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 6.62e5iT - 8.32e11T^{2} \)
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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

−10.05847991412702195472892582608, −9.808948522961568203185885785481, −8.574811115025335521030333970367, −7.51107620589345286352568475467, −6.73490976207107045688426908758, −5.75450595887221928627361773932, −4.21874245474464833553460549538, −3.25718268817466324560030063691, −2.02006410973201311219766655469, −0.77286646586765933622313190403, 0.881634623465041595269661740327, 2.45619738907895695481032876856, 3.21780395062585704034039461254, 4.69927767042003692480968567255, 5.66912474014578190673406905367, 6.68658125091188158875481298228, 7.987266871621462721646808540987, 8.819487169545528282209987899907, 9.674522923030929621093634925099, 10.29482020840205841022178929138

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