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Properties

Label 2-136-17.13-c1-0-3
Degree $2$
Conductor $136$
Sign $-0.788 + 0.615i$
Analytic cond. $1.08596$
Root an. cond. $1.04209$
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 − i)3-s + (−3 − 3i)5-s + (−3 + 3i)7-s − i·9-s + (1 − i)11-s + 2·13-s + 6i·15-s + (1 − 4i)17-s − 6i·19-s + 6·21-s + (1 − i)23-s + 13i·25-s + (−4 + 4i)27-s + (1 + i)29-s + (−1 − i)31-s + ⋯
L(s)  = 1  + (−0.577 − 0.577i)3-s + (−1.34 − 1.34i)5-s + (−1.13 + 1.13i)7-s − 0.333i·9-s + (0.301 − 0.301i)11-s + 0.554·13-s + 1.54i·15-s + (0.242 − 0.970i)17-s − 1.37i·19-s + 1.30·21-s + (0.208 − 0.208i)23-s + 2.60i·25-s + (−0.769 + 0.769i)27-s + (0.185 + 0.185i)29-s + (−0.179 − 0.179i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $-0.788 + 0.615i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ -0.788 + 0.615i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.159224 - 0.462660i\)
\(L(\frac12)\) \(\approx\) \(0.159224 - 0.462660i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
17 \( 1 + (-1 + 4i)T \)
good3 \( 1 + (1 + i)T + 3iT^{2} \)
5 \( 1 + (3 + 3i)T + 5iT^{2} \)
7 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 + (-1 + i)T - 11iT^{2} \)
13 \( 1 - 2T + 13T^{2} \)
19 \( 1 + 6iT - 19T^{2} \)
23 \( 1 + (-1 + i)T - 23iT^{2} \)
29 \( 1 + (-1 - i)T + 29iT^{2} \)
31 \( 1 + (1 + i)T + 31iT^{2} \)
37 \( 1 + (3 + 3i)T + 37iT^{2} \)
41 \( 1 + (-1 + i)T - 41iT^{2} \)
43 \( 1 - 2iT - 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 6iT - 59T^{2} \)
61 \( 1 + (-1 + i)T - 61iT^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 + (5 + 5i)T + 71iT^{2} \)
73 \( 1 + (-1 - i)T + 73iT^{2} \)
79 \( 1 + (-9 + 9i)T - 79iT^{2} \)
83 \( 1 + 14iT - 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 + (-13 - 13i)T + 97iT^{2} \)
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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

−12.59556677850495166800387194345, −11.97051129764903560079559465174, −11.31494624200146311970892108287, −9.202507382653078951283222717035, −8.854843738420638136300929156248, −7.36939679385969926518453818079, −6.19492569190470344880461988066, −4.98280709685367234778156530488, −3.38791501966806882203789319927, −0.54099548438271655965028962282, 3.48480366989158766057115751419, 4.10703016265840933503824180951, 6.15322450586680848039117927547, 7.11515201242943145173795287105, 8.080179679553920263428034571344, 10.09143045403590978449096306444, 10.48398298727560462919490705387, 11.34998283212830154083112140063, 12.41116369979000795244311433793, 13.71249579319047191607397152491

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