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Properties

Label 2-608-19.18-c2-0-38
Degree $2$
Conductor $608$
Sign $-0.830 - 0.556i$
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  − 4.29i·3-s + 2.32·5-s + 2.95·7-s − 9.45·9-s − 19.4·11-s + 14.0i·13-s − 10.0i·15-s − 22.2·17-s + (10.5 − 15.7i)19-s − 12.7i·21-s − 41.7·23-s − 19.5·25-s + 1.94i·27-s + 22.9i·29-s − 30.9i·31-s + ⋯
L(s)  = 1  − 1.43i·3-s + 0.465·5-s + 0.422·7-s − 1.05·9-s − 1.77·11-s + 1.07i·13-s − 0.666i·15-s − 1.31·17-s + (0.556 − 0.830i)19-s − 0.604i·21-s − 1.81·23-s − 0.783·25-s + 0.0722i·27-s + 0.790i·29-s − 0.999i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 608 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.830 - 0.556i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(608\)    =    \(2^{5} \cdot 19\)
Sign: $-0.830 - 0.556i$
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: \((2,\ 608,\ (\ :1),\ -0.830 - 0.556i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.5194391876\)
\(L(\frac12)\) \(\approx\) \(0.5194391876\)
\(L(2)\) 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 \( 1 + (-10.5 + 15.7i)T \)
good3 \( 1 + 4.29iT - 9T^{2} \)
5 \( 1 - 2.32T + 25T^{2} \)
7 \( 1 - 2.95T + 49T^{2} \)
11 \( 1 + 19.4T + 121T^{2} \)
13 \( 1 - 14.0iT - 169T^{2} \)
17 \( 1 + 22.2T + 289T^{2} \)
23 \( 1 + 41.7T + 529T^{2} \)
29 \( 1 - 22.9iT - 841T^{2} \)
31 \( 1 + 30.9iT - 961T^{2} \)
37 \( 1 + 62.1iT - 1.36e3T^{2} \)
41 \( 1 - 20.2iT - 1.68e3T^{2} \)
43 \( 1 - 53.2T + 1.84e3T^{2} \)
47 \( 1 + 21.3T + 2.20e3T^{2} \)
53 \( 1 + 12.4iT - 2.80e3T^{2} \)
59 \( 1 - 19.2iT - 3.48e3T^{2} \)
61 \( 1 - 4.96T + 3.72e3T^{2} \)
67 \( 1 - 58.9iT - 4.48e3T^{2} \)
71 \( 1 - 61.6iT - 5.04e3T^{2} \)
73 \( 1 - 14.5T + 5.32e3T^{2} \)
79 \( 1 + 113. iT - 6.24e3T^{2} \)
83 \( 1 - 48.9T + 6.88e3T^{2} \)
89 \( 1 + 27.8iT - 7.92e3T^{2} \)
97 \( 1 - 144. iT - 9.40e3T^{2} \)
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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

−9.904428688380996603871340093281, −8.890744586007928698907328828979, −7.889363231079611169549174746735, −7.36922674284757771654177650408, −6.37245767290389180415950824307, −5.54705679773189289976984677395, −4.34761017312171444003012484642, −2.42281347674840900481522449595, −1.92926327945013775636984002299, −0.17265900248403466814208497070, 2.22264260614315657648917682507, 3.40536091738980733712858919388, 4.56953475925931473890076636094, 5.30275818886988848323609005295, 6.08329048141029533783325278587, 7.79628815666360117748122926689, 8.287297629380430924615503731807, 9.525866174382483346167757648546, 10.22836373014059934466676983016, 10.55095650436336487473375410820

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