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Properties

Label 2-608-19.18-c2-0-33
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

\(L(\frac{3}{2})\) \(\approx\) \(0.07404911020\)
\(L(\frac12)\) \(\approx\) \(0.07404911020\)
\(L(2)\) 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

−10.10676150595722745038125845117, −9.592277052197316451275030374307, −8.850129197368362386228432510266, −7.61793799713175990189681935124, −6.06298918524970703906997095949, −5.82327971750799102292914013852, −4.36522215873531274357137105213, −3.60556473027606413037309749718, −2.53207935183094388489401632799, −0.02584898643312428507093851756, 1.57394529696449069364929687802, 2.57129716383923805108056065299, 3.86782250277526619141314511301, 5.67662350563398879255839366870, 6.38528595802182009381516845220, 6.88051497002834809765055939942, 7.903486958585285368882130686741, 8.904097390154684316121174269710, 9.775190099143370870822362992807, 10.58193545375830033976647633188

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