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Properties

Label 2-608-76.11-c0-0-0
Degree $2$
Conductor $608$
Sign $-0.0489 + 0.998i$
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

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

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.0489 + 0.998i$
Analytic conductor: \(0.303431\)
Root analytic conductor: \(0.550846\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (543, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :0),\ -0.0489 + 0.998i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6908813879\)
\(L(\frac12)\) \(\approx\) \(0.6908813879\)
\(L(1)\) 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

−10.83205702910891050293161287767, −9.696283833855589425937024717275, −8.973785578206036004416527972622, −7.968856433519597690605333638530, −6.91479889423352608118592797952, −5.96510884259150119521028902577, −5.32969246559001744155013287283, −4.27624416230100452575007135126, −2.54060602965108863061103159857, −0.921356769282575708589389190251, 2.12173342786610617706684343655, 3.46141081493330175932817934696, 4.84145422907222959210007397475, 5.49755724781655768682368337989, 6.65820829769792790057472118000, 7.17375748317411581951368264135, 8.624188539957715009766036029051, 9.578421417878958843292202680504, 10.39592739231545705204098948405, 10.93386315808261828901728721163

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