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Properties

Label 2-336-7.5-c6-0-17
Degree $2$
Conductor $336$
Sign $0.971 - 0.238i$
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 + (−160. − 92.6i)5-s + (338. + 53.3i)7-s + (121.5 − 210. i)9-s + (9.99e2 + 1.73e3i)11-s − 1.99e3i·13-s − 2.88e3·15-s + (−2.44e3 + 1.41e3i)17-s + (5.35e3 + 3.09e3i)19-s + (4.98e3 − 1.92e3i)21-s + (566. − 981. i)23-s + (9.35e3 + 1.62e4i)25-s − 3.78e3i·27-s − 3.12e4·29-s + (−4.67e4 + 2.69e4i)31-s + ⋯
L(s)  = 1  + (0.5 − 0.288i)3-s + (−1.28 − 0.741i)5-s + (0.987 + 0.155i)7-s + (0.166 − 0.288i)9-s + (0.751 + 1.30i)11-s − 0.909i·13-s − 0.855·15-s + (−0.497 + 0.287i)17-s + (0.780 + 0.450i)19-s + (0.538 − 0.207i)21-s + (0.0465 − 0.0806i)23-s + (0.598 + 1.03i)25-s − 0.192i·27-s − 1.27·29-s + (−1.56 + 0.905i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.971 - 0.238i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.971 - 0.238i$
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: \((2,\ 336,\ (\ :3),\ 0.971 - 0.238i)\)

Particular Values

\(L(\frac{7}{2})\) \(\approx\) \(2.151839552\)
\(L(\frac12)\) \(\approx\) \(2.151839552\)
\(L(4)\) 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 \( 1 + (-338. - 53.3i)T \)
good5 \( 1 + (160. + 92.6i)T + (7.81e3 + 1.35e4i)T^{2} \)
11 \( 1 + (-9.99e2 - 1.73e3i)T + (-8.85e5 + 1.53e6i)T^{2} \)
13 \( 1 + 1.99e3iT - 4.82e6T^{2} \)
17 \( 1 + (2.44e3 - 1.41e3i)T + (1.20e7 - 2.09e7i)T^{2} \)
19 \( 1 + (-5.35e3 - 3.09e3i)T + (2.35e7 + 4.07e7i)T^{2} \)
23 \( 1 + (-566. + 981. i)T + (-7.40e7 - 1.28e8i)T^{2} \)
29 \( 1 + 3.12e4T + 5.94e8T^{2} \)
31 \( 1 + (4.67e4 - 2.69e4i)T + (4.43e8 - 7.68e8i)T^{2} \)
37 \( 1 + (1.70e4 - 2.95e4i)T + (-1.28e9 - 2.22e9i)T^{2} \)
41 \( 1 - 1.25e4iT - 4.75e9T^{2} \)
43 \( 1 - 1.06e5T + 6.32e9T^{2} \)
47 \( 1 + (-1.19e5 - 6.87e4i)T + (5.38e9 + 9.33e9i)T^{2} \)
53 \( 1 + (-2.84e4 - 4.92e4i)T + (-1.10e10 + 1.91e10i)T^{2} \)
59 \( 1 + (-1.66e5 + 9.63e4i)T + (2.10e10 - 3.65e10i)T^{2} \)
61 \( 1 + (-3.27e5 - 1.89e5i)T + (2.57e10 + 4.46e10i)T^{2} \)
67 \( 1 + (-2.54e4 - 4.40e4i)T + (-4.52e10 + 7.83e10i)T^{2} \)
71 \( 1 + 3.79e5T + 1.28e11T^{2} \)
73 \( 1 + (-4.25e5 + 2.45e5i)T + (7.56e10 - 1.31e11i)T^{2} \)
79 \( 1 + (-4.45e5 + 7.71e5i)T + (-1.21e11 - 2.10e11i)T^{2} \)
83 \( 1 - 7.19e5iT - 3.26e11T^{2} \)
89 \( 1 + (-1.18e5 - 6.84e4i)T + (2.48e11 + 4.30e11i)T^{2} \)
97 \( 1 + 5.02e5iT - 8.32e11T^{2} \)
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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

−10.70527177170767879806258734474, −9.346441530567683117204004119713, −8.600817659782264781063046632133, −7.68528491460870511902342561445, −7.23231217633908787948247068376, −5.46105463520563514249439920059, −4.43299171556892174655813976531, −3.62619604571249231810167647159, −1.96330902270774959057642079827, −0.934995974186373680520058472483, 0.59766562284818825461554627608, 2.18763801068209969581079431304, 3.64086236287147775938863109524, 4.05642661160670031785097321790, 5.49702079910252326879406321235, 7.02520827648332105062725660360, 7.59463132087952446014081142936, 8.647028581435735669642746577924, 9.316176089152203587297589241777, 10.99089898753872076790681343261

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