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Properties

Label 2-608-19.18-c2-0-32
Degree $2$
Conductor $608$
Sign $-0.192 + 0.981i$
Analytic cond. $16.5668$
Root an. cond. $4.07023$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 4.61i·3-s + 3.04·5-s + 10.5·7-s − 12.2·9-s − 0.744·11-s − 9.55i·13-s − 14.0i·15-s + 9.60·17-s + (18.6 + 3.65i)19-s − 48.7i·21-s + 33.8·23-s − 15.7·25-s + 15.2i·27-s − 14.7i·29-s + 32.7i·31-s + ⋯
L(s)  = 1  − 1.53i·3-s + 0.608·5-s + 1.50·7-s − 1.36·9-s − 0.0677·11-s − 0.734i·13-s − 0.936i·15-s + 0.564·17-s + (0.981 + 0.192i)19-s − 2.32i·21-s + 1.47·23-s − 0.629·25-s + 0.563i·27-s − 0.509i·29-s + 1.05i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.192 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.192 + 0.981i$
Analytic conductor: \(16.5668\)
Root analytic conductor: \(4.07023\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{608} (417, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 608,\ (\ :1),\ -0.192 + 0.981i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.427153534\)
\(L(\frac12)\) \(\approx\) \(2.427153534\)
\(L(2)\) 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 \( 1 + (-18.6 - 3.65i)T \)
good3 \( 1 + 4.61iT - 9T^{2} \)
5 \( 1 - 3.04T + 25T^{2} \)
7 \( 1 - 10.5T + 49T^{2} \)
11 \( 1 + 0.744T + 121T^{2} \)
13 \( 1 + 9.55iT - 169T^{2} \)
17 \( 1 - 9.60T + 289T^{2} \)
23 \( 1 - 33.8T + 529T^{2} \)
29 \( 1 + 14.7iT - 841T^{2} \)
31 \( 1 - 32.7iT - 961T^{2} \)
37 \( 1 + 26.3iT - 1.36e3T^{2} \)
41 \( 1 - 35.4iT - 1.68e3T^{2} \)
43 \( 1 - 6.65T + 1.84e3T^{2} \)
47 \( 1 + 83.1T + 2.20e3T^{2} \)
53 \( 1 + 1.90iT - 2.80e3T^{2} \)
59 \( 1 + 86.1iT - 3.48e3T^{2} \)
61 \( 1 + 81.2T + 3.72e3T^{2} \)
67 \( 1 - 76.4iT - 4.48e3T^{2} \)
71 \( 1 + 72.4iT - 5.04e3T^{2} \)
73 \( 1 + 49.5T + 5.32e3T^{2} \)
79 \( 1 - 83.0iT - 6.24e3T^{2} \)
83 \( 1 - 22.0T + 6.88e3T^{2} \)
89 \( 1 + 143. iT - 7.92e3T^{2} \)
97 \( 1 - 27.2iT - 9.40e3T^{2} \)
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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

−10.30804417374842652362752981019, −9.149572296924673749007062262052, −8.002620815028145269873041679290, −7.74635508698598604076187211097, −6.70918140573588747168010782655, −5.65073403741238347398373725209, −4.95063674892070121051266534173, −3.04954134012859019852946825884, −1.80218344395742809977584670433, −1.05200726043608828051780430274, 1.56879089235481287603791972171, 3.09170563102569004632874196053, 4.34080493174500920697240383434, 5.01515250621139395730506455332, 5.72044874885207927337160087197, 7.22726384596286434980114617275, 8.280933076230250899603769810520, 9.209315294889198122804138789342, 9.733117439297282650156616020615, 10.69548892920890573325479390987

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