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Properties

Label 2-536-536.291-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.971 + 0.236i$
Analytic cond. $0.267498$
Root an. cond. $0.517202$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.235 + 0.971i)2-s + (−0.279 + 1.94i)3-s + (−0.888 + 0.458i)4-s + (−1.95 + 0.186i)6-s + (−0.654 − 0.755i)8-s + (−2.74 − 0.804i)9-s + (1.30 + 0.124i)11-s + (−0.642 − 1.85i)12-s + (0.580 − 0.814i)16-s + (−0.0845 − 0.0436i)17-s + (0.135 − 2.85i)18-s + (0.839 + 0.800i)19-s + (0.186 + 1.29i)22-s + (1.65 − 1.06i)24-s + (−0.654 + 0.755i)25-s + ⋯
L(s)  = 1  + (0.235 + 0.971i)2-s + (−0.279 + 1.94i)3-s + (−0.888 + 0.458i)4-s + (−1.95 + 0.186i)6-s + (−0.654 − 0.755i)8-s + (−2.74 − 0.804i)9-s + (1.30 + 0.124i)11-s + (−0.642 − 1.85i)12-s + (0.580 − 0.814i)16-s + (−0.0845 − 0.0436i)17-s + (0.135 − 2.85i)18-s + (0.839 + 0.800i)19-s + (0.186 + 1.29i)22-s + (1.65 − 1.06i)24-s + (−0.654 + 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.971 + 0.236i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.971 + 0.236i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (291, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.971 + 0.236i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8043711730\)
\(L(\frac12)\) \(\approx\) \(0.8043711730\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.235 - 0.971i)T \)
67 \( 1 + (-0.415 - 0.909i)T \)
good3 \( 1 + (0.279 - 1.94i)T + (-0.959 - 0.281i)T^{2} \)
5 \( 1 + (0.654 - 0.755i)T^{2} \)
7 \( 1 + (-0.0475 + 0.998i)T^{2} \)
11 \( 1 + (-1.30 - 0.124i)T + (0.981 + 0.189i)T^{2} \)
13 \( 1 + (-0.928 - 0.371i)T^{2} \)
17 \( 1 + (0.0845 + 0.0436i)T + (0.580 + 0.814i)T^{2} \)
19 \( 1 + (-0.839 - 0.800i)T + (0.0475 + 0.998i)T^{2} \)
23 \( 1 + (-0.723 + 0.690i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.928 + 0.371i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.0913 + 1.91i)T + (-0.995 + 0.0950i)T^{2} \)
43 \( 1 + (1.67 - 1.07i)T + (0.415 - 0.909i)T^{2} \)
47 \( 1 + (-0.235 - 0.971i)T^{2} \)
53 \( 1 + (-0.415 - 0.909i)T^{2} \)
59 \( 1 + (-0.428 - 0.494i)T + (-0.142 + 0.989i)T^{2} \)
61 \( 1 + (-0.981 + 0.189i)T^{2} \)
71 \( 1 + (-0.580 + 0.814i)T^{2} \)
73 \( 1 + (-1.76 + 0.168i)T + (0.981 - 0.189i)T^{2} \)
79 \( 1 + (0.786 - 0.618i)T^{2} \)
83 \( 1 + (-0.481 + 0.676i)T + (-0.327 - 0.945i)T^{2} \)
89 \( 1 + (0.0671 + 0.466i)T + (-0.959 + 0.281i)T^{2} \)
97 \( 1 + (-0.888 - 1.53i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.56431820686998927207169161491, −10.33753471279326486831438626810, −9.577619487340299252839137044770, −9.070169054287882617387025720190, −8.142480918114329503532709585209, −6.75143107717264884633779618868, −5.74151826131807731753915501185, −5.05635225904086100363716887548, −3.97213527325776573242327846751, −3.47338801847241784335548658305, 1.05623195569483194404743320644, 2.15033179344780173187865113960, 3.35656841721294082923895781900, 4.97016355779598015456895290200, 6.10935037686342701930285492382, 6.76591640713631632486557223966, 7.935951896229082691259153192770, 8.739058354592761239913953812753, 9.702067721775972878379868462856, 11.12638268502446788351675603275

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