Properties

Label 2-536-536.19-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.128 - 0.991i$
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  + (−0.888 − 0.458i)2-s + (−1.78 + 0.523i)3-s + (0.580 + 0.814i)4-s + (1.82 + 0.351i)6-s + (−0.142 − 0.989i)8-s + (2.05 − 1.32i)9-s + (−0.279 + 0.0538i)11-s + (−1.45 − 1.14i)12-s + (−0.327 + 0.945i)16-s + (−1.15 + 1.62i)17-s + (−2.43 + 0.232i)18-s + (−0.0311 + 0.653i)19-s + (0.273 + 0.0801i)22-s + (0.771 + 1.68i)24-s + (−0.142 + 0.989i)25-s + ⋯
L(s)  = 1  + (−0.888 − 0.458i)2-s + (−1.78 + 0.523i)3-s + (0.580 + 0.814i)4-s + (1.82 + 0.351i)6-s + (−0.142 − 0.989i)8-s + (2.05 − 1.32i)9-s + (−0.279 + 0.0538i)11-s + (−1.45 − 1.14i)12-s + (−0.327 + 0.945i)16-s + (−1.15 + 1.62i)17-s + (−2.43 + 0.232i)18-s + (−0.0311 + 0.653i)19-s + (0.273 + 0.0801i)22-s + (0.771 + 1.68i)24-s + (−0.142 + 0.989i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.128 - 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.128 - 0.991i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.128 - 0.991i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2327720409\)
\(L(\frac12)\) \(\approx\) \(0.2327720409\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (0.654 + 0.755i)T \)
good3 \( 1 + (1.78 - 0.523i)T + (0.841 - 0.540i)T^{2} \)
5 \( 1 + (0.142 - 0.989i)T^{2} \)
7 \( 1 + (0.995 - 0.0950i)T^{2} \)
11 \( 1 + (0.279 - 0.0538i)T + (0.928 - 0.371i)T^{2} \)
13 \( 1 + (-0.723 + 0.690i)T^{2} \)
17 \( 1 + (1.15 - 1.62i)T + (-0.327 - 0.945i)T^{2} \)
19 \( 1 + (0.0311 - 0.653i)T + (-0.995 - 0.0950i)T^{2} \)
23 \( 1 + (-0.0475 - 0.998i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.723 - 0.690i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1.67 + 0.159i)T + (0.981 + 0.189i)T^{2} \)
43 \( 1 + (-0.815 - 1.78i)T + (-0.654 + 0.755i)T^{2} \)
47 \( 1 + (0.888 + 0.458i)T^{2} \)
53 \( 1 + (0.654 + 0.755i)T^{2} \)
59 \( 1 + (-0.223 - 1.55i)T + (-0.959 + 0.281i)T^{2} \)
61 \( 1 + (-0.928 - 0.371i)T^{2} \)
71 \( 1 + (0.327 - 0.945i)T^{2} \)
73 \( 1 + (-1.13 - 0.219i)T + (0.928 + 0.371i)T^{2} \)
79 \( 1 + (-0.235 - 0.971i)T^{2} \)
83 \( 1 + (-0.428 + 1.23i)T + (-0.786 - 0.618i)T^{2} \)
89 \( 1 + (-1.70 - 0.500i)T + (0.841 + 0.540i)T^{2} \)
97 \( 1 + (0.580 + 1.00i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.09675696462078402431006090612, −10.52470554175185386323086844153, −9.877935173635781781221812342200, −8.873125329180252710460223630306, −7.70628298510694650168306910083, −6.58698637480964494955441779372, −5.96456314938677455397162068902, −4.67330298658023892868892142430, −3.66148928898336267019641689529, −1.60152759867534834100729695222, 0.47191492857109870696145206375, 2.18691422719419511696280571902, 4.76927632797476914234002286864, 5.37313966981860141153294533635, 6.56360168857823494932353804723, 6.88591959102589787791584610036, 7.894377574125656567423529807439, 9.100877991473546013658623684979, 10.10644353909955643044276118020, 10.84118053678599994159358043525

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