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Properties

Label 2-136-17.16-c1-0-3
Degree $2$
Conductor $136$
Sign $0.242 + 0.970i$
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  − 2i·3-s − 2i·7-s − 9-s − 2i·11-s − 2·13-s + (1 + 4i)17-s + 4·19-s − 4·21-s + 6i·23-s + 5·25-s − 4i·27-s + 8i·29-s + 6i·31-s − 4·33-s − 8i·37-s + ⋯
L(s)  = 1  − 1.15i·3-s − 0.755i·7-s − 0.333·9-s − 0.603i·11-s − 0.554·13-s + (0.242 + 0.970i)17-s + 0.917·19-s − 0.872·21-s + 1.25i·23-s + 25-s − 0.769i·27-s + 1.48i·29-s + 1.07i·31-s − 0.696·33-s − 1.31i·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 136 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(136\)    =    \(2^{3} \cdot 17\)
Sign: $0.242 + 0.970i$
Analytic conductor: \(1.08596\)
Root analytic conductor: \(1.04209\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{136} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 136,\ (\ :1/2),\ 0.242 + 0.970i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.859018 - 0.670701i\)
\(L(\frac12)\) \(\approx\) \(0.859018 - 0.670701i\)
\(L(\frac{3}{2})\) 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 \( 1 + 2iT - 3T^{2} \)
5 \( 1 - 5T^{2} \)
7 \( 1 + 2iT - 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 6iT - 23T^{2} \)
29 \( 1 - 8iT - 29T^{2} \)
31 \( 1 - 6iT - 31T^{2} \)
37 \( 1 + 8iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 8T + 47T^{2} \)
53 \( 1 - 6T + 53T^{2} \)
59 \( 1 - 4T + 59T^{2} \)
61 \( 1 + 8iT - 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + 8iT - 73T^{2} \)
79 \( 1 + 2iT - 79T^{2} \)
83 \( 1 + 4T + 83T^{2} \)
89 \( 1 + 14T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−12.99970093619350540574691089073, −12.21243800068672831929013792633, −11.07242025397519477334991724911, −9.995020016740050902212524492229, −8.557391111464208381852524572871, −7.45396792800417318047712352702, −6.75519756916216776471804371149, −5.29373585007761846360158851190, −3.43152521977650104888811487447, −1.38802024822044773221649077424, 2.78019940817490253977477595156, 4.44533820100233128512511544231, 5.32996690938990440941317947043, 6.93225600075929577769235561086, 8.375826370607383914815799145297, 9.620985704708839645840040194600, 10.01145457652993942184481535222, 11.41856612633389145788721788846, 12.22589104631522220457841880689, 13.48035461708353451476303724663

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