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Properties

Label 2-608-152.75-c1-0-15
Degree $2$
Conductor $608$
Sign $-0.718 + 0.695i$
Analytic cond. $4.85490$
Root an. cond. $2.20338$
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  − 1.26i·3-s − 3.51i·5-s − 2.23i·7-s + 1.40·9-s − 2.89·11-s + 6.30·13-s − 4.43·15-s − 4.79·17-s + (−0.895 + 4.26i)19-s − 2.82·21-s − 0.524i·23-s − 7.38·25-s − 5.56i·27-s − 0.415·29-s + 1.20·31-s + ⋯
L(s)  = 1  − 0.728i·3-s − 1.57i·5-s − 0.845i·7-s + 0.469·9-s − 0.872·11-s + 1.74·13-s − 1.14·15-s − 1.16·17-s + (−0.205 + 0.978i)19-s − 0.615·21-s − 0.109i·23-s − 1.47·25-s − 1.07i·27-s − 0.0771·29-s + 0.216·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.718 + 0.695i$
Analytic conductor: \(4.85490\)
Root analytic conductor: \(2.20338\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1/2),\ -0.718 + 0.695i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.528986 - 1.30737i\)
\(L(\frac12)\) \(\approx\) \(0.528986 - 1.30737i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 + (0.895 - 4.26i)T \)
good3 \( 1 + 1.26iT - 3T^{2} \)
5 \( 1 + 3.51iT - 5T^{2} \)
7 \( 1 + 2.23iT - 7T^{2} \)
11 \( 1 + 2.89T + 11T^{2} \)
13 \( 1 - 6.30T + 13T^{2} \)
17 \( 1 + 4.79T + 17T^{2} \)
23 \( 1 + 0.524iT - 23T^{2} \)
29 \( 1 + 0.415T + 29T^{2} \)
31 \( 1 - 1.20T + 31T^{2} \)
37 \( 1 + 5.88T + 37T^{2} \)
41 \( 1 - 4.87iT - 41T^{2} \)
43 \( 1 - 10.6T + 43T^{2} \)
47 \( 1 + 4.27iT - 47T^{2} \)
53 \( 1 + 6.05T + 53T^{2} \)
59 \( 1 - 8.08iT - 59T^{2} \)
61 \( 1 + 8.38iT - 61T^{2} \)
67 \( 1 + 9.79iT - 67T^{2} \)
71 \( 1 - 10.2T + 71T^{2} \)
73 \( 1 - 7.76T + 73T^{2} \)
79 \( 1 + 7.33T + 79T^{2} \)
83 \( 1 + 2T + 83T^{2} \)
89 \( 1 - 12.1iT - 89T^{2} \)
97 \( 1 + 2.19iT - 97T^{2} \)
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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

−10.40747680570945906319193670074, −9.286180557192045611825125511429, −8.372622967464181823661299329271, −7.88354128251178904104183671002, −6.73864148902304733726370301043, −5.79483852975750999474009944853, −4.62203039216079875259067480429, −3.84268292788005262895035249660, −1.81796771242166068759583685078, −0.820865092014398801765497966800, 2.27663457949852434079739106219, 3.26239094933155171026850961066, 4.29436123806502571418541421781, 5.58328188334030538676973716551, 6.48729933519376658203238532039, 7.27585229322549331259317349233, 8.527072621161898494582525014887, 9.268082570501736604061050453712, 10.35719039214751599298517757200, 10.92146458018460582844574763435

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