Properties

Label 2-608-152.131-c0-0-0
Degree $2$
Conductor $608$
Sign $0.755 + 0.654i$
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  + (−0.266 − 0.223i)3-s + (−0.152 − 0.866i)9-s + (0.766 − 1.32i)11-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)19-s + (0.766 − 0.642i)25-s + (−0.326 + 0.565i)27-s + (−0.5 + 0.181i)33-s + (−1.43 − 1.20i)41-s + (−0.939 + 0.342i)43-s + (−0.5 + 0.866i)49-s + (0.266 − 0.223i)51-s + (−0.173 − 0.300i)57-s + (−0.266 + 1.50i)59-s + (0.326 + 1.85i)67-s + ⋯
L(s)  = 1  + (−0.266 − 0.223i)3-s + (−0.152 − 0.866i)9-s + (0.766 − 1.32i)11-s + (−0.173 + 0.984i)17-s + (0.939 + 0.342i)19-s + (0.766 − 0.642i)25-s + (−0.326 + 0.565i)27-s + (−0.5 + 0.181i)33-s + (−1.43 − 1.20i)41-s + (−0.939 + 0.342i)43-s + (−0.5 + 0.866i)49-s + (0.266 − 0.223i)51-s + (−0.173 − 0.300i)57-s + (−0.266 + 1.50i)59-s + (0.326 + 1.85i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.654i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $0.755 + 0.654i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (207, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :0),\ 0.755 + 0.654i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8640133113\)
\(L(\frac12)\) \(\approx\) \(0.8640133113\)
\(L(1)\) 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 \( 1 + (-0.939 - 0.342i)T \)
good3 \( 1 + (0.266 + 0.223i)T + (0.173 + 0.984i)T^{2} \)
5 \( 1 + (-0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.766 + 1.32i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (-0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.173 - 0.984i)T + (-0.939 - 0.342i)T^{2} \)
23 \( 1 + (-0.766 - 0.642i)T^{2} \)
29 \( 1 + (0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (1.43 + 1.20i)T + (0.173 + 0.984i)T^{2} \)
43 \( 1 + (0.939 - 0.342i)T + (0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 - 0.342i)T^{2} \)
53 \( 1 + (-0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.266 - 1.50i)T + (-0.939 - 0.342i)T^{2} \)
61 \( 1 + (-0.766 - 0.642i)T^{2} \)
67 \( 1 + (-0.326 - 1.85i)T + (-0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 + 0.642i)T^{2} \)
73 \( 1 + (-1.17 - 0.984i)T + (0.173 + 0.984i)T^{2} \)
79 \( 1 + (-0.173 - 0.984i)T^{2} \)
83 \( 1 + (0.939 + 1.62i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.766 - 0.642i)T + (0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.326 - 1.85i)T + (-0.939 - 0.342i)T^{2} \)
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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

−10.86513780020610467402695864701, −9.905488152862682249298056440472, −8.876364346904785175355093423578, −8.331051208528418262941421363205, −7.00064296378201153496216777063, −6.24584583255770055255394976602, −5.48808465552932824849095915287, −3.99397443482827874161638752104, −3.13031585724353341521928791601, −1.22935428165334073002954750685, 1.83779805343509708672607882732, 3.25986610755898420779037744957, 4.73310911845416777839096337163, 5.13954695439333345400622437016, 6.63185417040064182495879095091, 7.29498620924618147029399693010, 8.318044661630414524699344578636, 9.479172007547607425185062254661, 9.921827146520219298617441610317, 11.08294334497560413895395610556

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