Properties

Label 2-536-536.451-c0-0-0
Degree $2$
Conductor $536$
Sign $0.791 + 0.610i$
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  + (0.580 − 0.814i)2-s + (1.21 + 0.782i)3-s + (−0.327 − 0.945i)4-s + (1.34 − 0.537i)6-s + (−0.959 − 0.281i)8-s + (0.454 + 0.996i)9-s + (−1.78 − 0.713i)11-s + (0.341 − 1.40i)12-s + (−0.786 + 0.618i)16-s + (−0.642 + 1.85i)17-s + (1.07 + 0.207i)18-s + (1.56 − 0.149i)19-s + (−1.61 + 1.03i)22-s + (−0.947 − 1.09i)24-s + (−0.959 + 0.281i)25-s + ⋯
L(s)  = 1  + (0.580 − 0.814i)2-s + (1.21 + 0.782i)3-s + (−0.327 − 0.945i)4-s + (1.34 − 0.537i)6-s + (−0.959 − 0.281i)8-s + (0.454 + 0.996i)9-s + (−1.78 − 0.713i)11-s + (0.341 − 1.40i)12-s + (−0.786 + 0.618i)16-s + (−0.642 + 1.85i)17-s + (1.07 + 0.207i)18-s + (1.56 − 0.149i)19-s + (−1.61 + 1.03i)22-s + (−0.947 − 1.09i)24-s + (−0.959 + 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.791 + 0.610i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.791 + 0.610i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (451, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.791 + 0.610i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.478652399\)
\(L(\frac12)\) \(\approx\) \(1.478652399\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.580 + 0.814i)T \)
67 \( 1 + (0.142 + 0.989i)T \)
good3 \( 1 + (-1.21 - 0.782i)T + (0.415 + 0.909i)T^{2} \)
5 \( 1 + (0.959 - 0.281i)T^{2} \)
7 \( 1 + (-0.981 - 0.189i)T^{2} \)
11 \( 1 + (1.78 + 0.713i)T + (0.723 + 0.690i)T^{2} \)
13 \( 1 + (-0.0475 - 0.998i)T^{2} \)
17 \( 1 + (0.642 - 1.85i)T + (-0.786 - 0.618i)T^{2} \)
19 \( 1 + (-1.56 + 0.149i)T + (0.981 - 0.189i)T^{2} \)
23 \( 1 + (0.995 + 0.0950i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.0475 + 0.998i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-0.815 + 0.157i)T + (0.928 - 0.371i)T^{2} \)
43 \( 1 + (1.21 + 1.40i)T + (-0.142 + 0.989i)T^{2} \)
47 \( 1 + (-0.580 + 0.814i)T^{2} \)
53 \( 1 + (0.142 + 0.989i)T^{2} \)
59 \( 1 + (0.452 + 0.132i)T + (0.841 + 0.540i)T^{2} \)
61 \( 1 + (-0.723 + 0.690i)T^{2} \)
71 \( 1 + (0.786 - 0.618i)T^{2} \)
73 \( 1 + (0.607 - 0.243i)T + (0.723 - 0.690i)T^{2} \)
79 \( 1 + (0.888 + 0.458i)T^{2} \)
83 \( 1 + (-0.223 + 0.175i)T + (0.235 - 0.971i)T^{2} \)
89 \( 1 + (-0.975 + 0.627i)T + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (-0.327 - 0.566i)T + (-0.5 + 0.866i)T^{2} \)
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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

−10.64818561497387610850543707682, −10.29174606137008409426233585716, −9.320511001884030618226261198859, −8.519028400605846676359316052458, −7.69951364862237916591459558775, −5.95914541641173437228640507714, −5.08196379787078038861567377956, −3.89505468012751215367513600783, −3.17360232788603542944038323618, −2.15532631033475683501512479779, 2.44789873355177285875543061234, 3.10238042997708542561328001586, 4.66238773278856029964343142357, 5.52837254848736888080149057949, 6.97705294189704329659550628292, 7.57404851001099445438516457808, 8.024937487165289866748641065495, 9.140081146205538347023455814631, 9.876289222433458471499657649335, 11.45762410095587770807193194690

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