Properties

Label 2-608-152.43-c0-0-0
Degree $2$
Conductor $608$
Sign $-0.305 + 0.952i$
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  + (0.326 − 1.85i)3-s + (−2.37 − 0.866i)9-s + (0.173 − 0.300i)11-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)19-s + (0.173 + 0.984i)25-s + (−1.43 + 2.49i)27-s + (−0.5 − 0.419i)33-s + (0.266 − 1.50i)41-s + (0.766 + 0.642i)43-s + (−0.5 + 0.866i)49-s + (−0.326 − 1.85i)51-s + (0.939 + 1.62i)57-s + (0.326 − 0.118i)59-s + (1.43 + 0.524i)67-s + ⋯
L(s)  = 1  + (0.326 − 1.85i)3-s + (−2.37 − 0.866i)9-s + (0.173 − 0.300i)11-s + (0.939 − 0.342i)17-s + (−0.766 + 0.642i)19-s + (0.173 + 0.984i)25-s + (−1.43 + 2.49i)27-s + (−0.5 − 0.419i)33-s + (0.266 − 1.50i)41-s + (0.766 + 0.642i)43-s + (−0.5 + 0.866i)49-s + (−0.326 − 1.85i)51-s + (0.939 + 1.62i)57-s + (0.326 − 0.118i)59-s + (1.43 + 0.524i)67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.305 + 0.952i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.305 + 0.952i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (271, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :0),\ -0.305 + 0.952i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9705197867\)
\(L(\frac12)\) \(\approx\) \(0.9705197867\)
\(L(1)\) 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 \( 1 + (0.766 - 0.642i)T \)
good3 \( 1 + (-0.326 + 1.85i)T + (-0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 - 0.866i)T^{2} \)
11 \( 1 + (-0.173 + 0.300i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.939 - 0.342i)T^{2} \)
17 \( 1 + (-0.939 + 0.342i)T + (0.766 - 0.642i)T^{2} \)
23 \( 1 + (-0.173 + 0.984i)T^{2} \)
29 \( 1 + (-0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (-0.266 + 1.50i)T + (-0.939 - 0.342i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T + (0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 - 0.642i)T^{2} \)
53 \( 1 + (-0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.326 + 0.118i)T + (0.766 - 0.642i)T^{2} \)
61 \( 1 + (-0.173 + 0.984i)T^{2} \)
67 \( 1 + (-1.43 - 0.524i)T + (0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 - 0.984i)T^{2} \)
73 \( 1 + (-0.0603 + 0.342i)T + (-0.939 - 0.342i)T^{2} \)
79 \( 1 + (0.939 + 0.342i)T^{2} \)
83 \( 1 + (-0.766 - 1.32i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
97 \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \)
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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

−10.84776614561509628390077135239, −9.494030069448359496065890732319, −8.585981900216871770114248886632, −7.83347785922591937469568122457, −7.14560303643530164893683453678, −6.24707139947005398655872817425, −5.44905249053804748627334394089, −3.60732893172876801000684977023, −2.45645617869853550690441822292, −1.23846374553900447444910465848, 2.57086109078144494480193404647, 3.68678506015844732674513004158, 4.50057578018201204885039798047, 5.34286322463509042510694121640, 6.44315153358105333983504348681, 7.952131748532381610905778392601, 8.693697641278356447360767733568, 9.539986994179437945453412388898, 10.16363206862486454347231373740, 10.84968925644618796413075463288

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