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Properties

Label 2-336-112.19-c1-0-19
Degree $2$
Conductor $336$
Sign $-0.220 + 0.975i$
Analytic cond. $2.68297$
Root an. cond. $1.63797$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.668 − 1.24i)2-s + (0.258 − 0.965i)3-s + (−1.10 + 1.66i)4-s + (0.513 − 0.137i)5-s + (−1.37 + 0.323i)6-s + (1.08 + 2.41i)7-s + (2.81 + 0.262i)8-s + (−0.866 − 0.499i)9-s + (−0.515 − 0.548i)10-s + (1.67 − 6.25i)11-s + (1.32 + 1.49i)12-s + (3.08 − 3.08i)13-s + (2.28 − 2.96i)14-s − 0.532i·15-s + (−1.55 − 3.68i)16-s + (−1.95 + 1.12i)17-s + ⋯
L(s)  = 1  + (−0.472 − 0.881i)2-s + (0.149 − 0.557i)3-s + (−0.552 + 0.833i)4-s + (0.229 − 0.0615i)5-s + (−0.562 + 0.132i)6-s + (0.408 + 0.912i)7-s + (0.995 + 0.0928i)8-s + (−0.288 − 0.166i)9-s + (−0.162 − 0.173i)10-s + (0.505 − 1.88i)11-s + (0.382 + 0.432i)12-s + (0.856 − 0.856i)13-s + (0.611 − 0.791i)14-s − 0.137i·15-s + (−0.389 − 0.921i)16-s + (−0.474 + 0.274i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.220 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.220 + 0.975i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.220 + 0.975i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.714248 - 0.893920i\)
\(L(\frac12)\) \(\approx\) \(0.714248 - 0.893920i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.668 + 1.24i)T \)
3 \( 1 + (-0.258 + 0.965i)T \)
7 \( 1 + (-1.08 - 2.41i)T \)
good5 \( 1 + (-0.513 + 0.137i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-1.67 + 6.25i)T + (-9.52 - 5.5i)T^{2} \)
13 \( 1 + (-3.08 + 3.08i)T - 13iT^{2} \)
17 \( 1 + (1.95 - 1.12i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.117 + 0.0316i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-2.16 + 3.74i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.37 - 4.37i)T + 29iT^{2} \)
31 \( 1 + (1.25 + 2.17i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-0.554 - 2.07i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 - 8.63T + 41T^{2} \)
43 \( 1 + (2.21 + 2.21i)T + 43iT^{2} \)
47 \( 1 + (-4.58 + 7.93i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (8.33 + 2.23i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (5.57 + 1.49i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-2.57 - 9.61i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (-7.44 - 1.99i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 3.75T + 71T^{2} \)
73 \( 1 + (-0.753 - 1.30i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (4.23 + 2.44i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-10.6 - 10.6i)T + 83iT^{2} \)
89 \( 1 + (5.81 - 10.0i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 15.5iT - 97T^{2} \)
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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

−11.23331673619879888926219838086, −10.66841010576075273548098773104, −9.178283984570928615816619562493, −8.592453477280786032844922139247, −7.999978840884719086843671449834, −6.38056872032684765412391161760, −5.40654968837449344076530077731, −3.65750718572303105256214515404, −2.59576287654804650322391531974, −1.08305427174569094004883287303, 1.69728468493281500679760318040, 4.17457324311869760746443160207, 4.67785890852066604499505743847, 6.20059400593866653500183494286, 7.11709481923910221648264838918, 7.928896270257604393880487374248, 9.219738069799160271561025329335, 9.679973241882918599951070524488, 10.62690350835940768353747898176, 11.51958377200594347713959470447

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