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Properties

Label 2-336-12.11-c1-0-2
Degree $2$
Conductor $336$
Sign $-0.138 - 0.990i$
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  + (1.06 + 1.36i)3-s + 2.12i·5-s + i·7-s + (−0.732 + 2.90i)9-s − 5.81·11-s + 4.19·13-s + (−2.90 + 2.26i)15-s − 5.81i·17-s + 2.73i·19-s + (−1.36 + 1.06i)21-s + 4.25·23-s + 0.464·25-s + (−4.75 + 2.09i)27-s + 5.81i·29-s − 2.53i·31-s + ⋯
L(s)  = 1  + (0.614 + 0.788i)3-s + 0.952i·5-s + 0.377i·7-s + (−0.244 + 0.969i)9-s − 1.75·11-s + 1.16·13-s + (−0.751 + 0.585i)15-s − 1.41i·17-s + 0.626i·19-s + (−0.298 + 0.232i)21-s + 0.888·23-s + 0.0928·25-s + (−0.914 + 0.403i)27-s + 1.08i·29-s − 0.455i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.138 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(336\)    =    \(2^{4} \cdot 3 \cdot 7\)
Sign: $-0.138 - 0.990i$
Analytic conductor: \(2.68297\)
Root analytic conductor: \(1.63797\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{336} (239, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 336,\ (\ :1/2),\ -0.138 - 0.990i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.974624 + 1.11995i\)
\(L(\frac12)\) \(\approx\) \(0.974624 + 1.11995i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.06 - 1.36i)T \)
7 \( 1 - iT \)
good5 \( 1 - 2.12iT - 5T^{2} \)
11 \( 1 + 5.81T + 11T^{2} \)
13 \( 1 - 4.19T + 13T^{2} \)
17 \( 1 + 5.81iT - 17T^{2} \)
19 \( 1 - 2.73iT - 19T^{2} \)
23 \( 1 - 4.25T + 23T^{2} \)
29 \( 1 - 5.81iT - 29T^{2} \)
31 \( 1 + 2.53iT - 31T^{2} \)
37 \( 1 - 11.4T + 37T^{2} \)
41 \( 1 - 1.55iT - 41T^{2} \)
43 \( 1 + 2iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 1.55iT - 53T^{2} \)
59 \( 1 - 9.50T + 59T^{2} \)
61 \( 1 + 1.26T + 61T^{2} \)
67 \( 1 + 3.46iT - 67T^{2} \)
71 \( 1 - 1.55T + 71T^{2} \)
73 \( 1 + 11.4T + 73T^{2} \)
79 \( 1 + 12iT - 79T^{2} \)
83 \( 1 - 9.50T + 83T^{2} \)
89 \( 1 + 13.1iT - 89T^{2} \)
97 \( 1 + 4.92T + 97T^{2} \)
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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

−11.39612676283031287405379733853, −10.80503151441046982365752610985, −10.06241129845367043666052447816, −9.057996381989589280495310556461, −8.092946925571855036536510520249, −7.21809898121271525566161047137, −5.78769038163167658983403266387, −4.80487561728565065761467431716, −3.27042541336623839488462742754, −2.58836993737451655143473234505, 1.05091088301444855998878843524, 2.64504144420863156286251031827, 4.07906452628281230306348856295, 5.42046890238469436342551259833, 6.50747177935121535829309302776, 7.81613694385771668658959688216, 8.307503022402787556248978611245, 9.174319908073314941749618396249, 10.41733811092582825487555820400, 11.31405911834868646773980467456

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