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Properties

Label 2-608-76.7-c0-0-1
Degree $2$
Conductor $608$
Sign $0.998 - 0.0489i$
Analytic cond. $0.303431$
Root an. cond. $0.550846$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)13-s + (0.866 + 0.499i)15-s + (−0.5 − 0.866i)17-s − i·19-s + (−0.866 − 0.5i)23-s + i·27-s + (−0.5 + 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 49-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)5-s + (−0.5 + 0.866i)13-s + (0.866 + 0.499i)15-s + (−0.5 − 0.866i)17-s − i·19-s + (−0.866 − 0.5i)23-s + i·27-s + (−0.5 + 0.866i)29-s − 2i·31-s + 0.999i·39-s + (−0.5 − 0.866i)41-s + (−0.866 + 0.5i)43-s + (0.866 + 0.5i)47-s + 49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0489i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0489i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.998 - 0.0489i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (159, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :0),\ 0.998 - 0.0489i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.213750997\)
\(L(\frac12)\) \(\approx\) \(1.213750997\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + iT \)
good3 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
17 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + 2iT - T^{2} \)
37 \( 1 + T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
47 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
53 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 - 2iT - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.91359272047073499292151406630, −9.829018210476068084258541442729, −9.145581742806983750272769408039, −8.230082576414842904199843995844, −7.15947266291594548559820984459, −6.74703446302938217338749664756, −5.44132731449757688771682563388, −4.14650064633458554788531923730, −2.68355143182558775553460465950, −2.18537959018974700229037322651, 1.78396047555416462650800385223, 3.18827094450451246200727851099, 4.16908640299428815677374098549, 5.30889111847725754477391362922, 6.15609306423635503235058732235, 7.60040728919371533297932533832, 8.480730532297277286426855379461, 8.991229190125188849710362563126, 9.976728757728069338142255299134, 10.42957138282685731552418635329

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