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Properties

Label 2-336-21.17-c3-0-30
Degree $2$
Conductor $336$
Sign $0.869 + 0.494i$
Analytic cond. $19.8246$
Root an. cond. $4.45248$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (5.16 + 0.550i)3-s + (−0.746 − 1.29i)5-s + (−15.9 − 9.49i)7-s + (26.3 + 5.69i)9-s + (38.1 + 22.0i)11-s − 63.3i·13-s + (−3.14 − 7.08i)15-s + (−55.7 + 96.5i)17-s + (109. − 63.3i)19-s + (−76.9 − 57.8i)21-s + (176. − 102. i)23-s + (61.3 − 106. i)25-s + (133. + 43.9i)27-s − 115. i·29-s + (74.9 + 43.2i)31-s + ⋯
L(s)  = 1  + (0.994 + 0.105i)3-s + (−0.0667 − 0.115i)5-s + (−0.858 − 0.512i)7-s + (0.977 + 0.210i)9-s + (1.04 + 0.604i)11-s − 1.35i·13-s + (−0.0541 − 0.122i)15-s + (−0.795 + 1.37i)17-s + (1.32 − 0.764i)19-s + (−0.799 − 0.600i)21-s + (1.60 − 0.925i)23-s + (0.491 − 0.850i)25-s + (0.949 + 0.313i)27-s − 0.738i·29-s + (0.434 + 0.250i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.869 + 0.494i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $0.869 + 0.494i$
Analytic conductor: \(19.8246\)
Root analytic conductor: \(4.45248\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :3/2),\ 0.869 + 0.494i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.624705667\)
\(L(\frac12)\) \(\approx\) \(2.624705667\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5.16 - 0.550i)T \)
7 \( 1 + (15.9 + 9.49i)T \)
good5 \( 1 + (0.746 + 1.29i)T + (-62.5 + 108. i)T^{2} \)
11 \( 1 + (-38.1 - 22.0i)T + (665.5 + 1.15e3i)T^{2} \)
13 \( 1 + 63.3iT - 2.19e3T^{2} \)
17 \( 1 + (55.7 - 96.5i)T + (-2.45e3 - 4.25e3i)T^{2} \)
19 \( 1 + (-109. + 63.3i)T + (3.42e3 - 5.94e3i)T^{2} \)
23 \( 1 + (-176. + 102. i)T + (6.08e3 - 1.05e4i)T^{2} \)
29 \( 1 + 115. iT - 2.43e4T^{2} \)
31 \( 1 + (-74.9 - 43.2i)T + (1.48e4 + 2.57e4i)T^{2} \)
37 \( 1 + (53.8 + 93.3i)T + (-2.53e4 + 4.38e4i)T^{2} \)
41 \( 1 + 336.T + 6.89e4T^{2} \)
43 \( 1 + 191.T + 7.95e4T^{2} \)
47 \( 1 + (-158. - 274. i)T + (-5.19e4 + 8.99e4i)T^{2} \)
53 \( 1 + (-321. - 185. i)T + (7.44e4 + 1.28e5i)T^{2} \)
59 \( 1 + (-187. + 325. i)T + (-1.02e5 - 1.77e5i)T^{2} \)
61 \( 1 + (-38.7 + 22.4i)T + (1.13e5 - 1.96e5i)T^{2} \)
67 \( 1 + (-207. + 359. i)T + (-1.50e5 - 2.60e5i)T^{2} \)
71 \( 1 - 676. iT - 3.57e5T^{2} \)
73 \( 1 + (587. + 339. i)T + (1.94e5 + 3.36e5i)T^{2} \)
79 \( 1 + (289. + 500. i)T + (-2.46e5 + 4.26e5i)T^{2} \)
83 \( 1 - 253.T + 5.71e5T^{2} \)
89 \( 1 + (-350. - 607. i)T + (-3.52e5 + 6.10e5i)T^{2} \)
97 \( 1 - 142. iT - 9.12e5T^{2} \)
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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

−10.74503791250280070884352541518, −10.05254814747147398589003820720, −9.141575691662966596028267438566, −8.366110263899458262682436209778, −7.20710623186531387047674410772, −6.47715142336165441202916840738, −4.77795496175636098915470093626, −3.71219575074979408328392172656, −2.72260911932250278967258255352, −0.982247025332154137590336050135, 1.37143940074296707333815633437, 2.93084790562054875669518665863, 3.69870309417636789630047529999, 5.18845223555439595829749834286, 6.79893692576227111449839039171, 7.10360933496690558838351421848, 8.802459014274487888921785028055, 9.127148327750662332458349177223, 9.897827126852595810108282767127, 11.48820879799183728993694662442

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