Properties

Label 2-536-536.419-c0-0-0
Degree $2$
Conductor $536$
Sign $0.982 - 0.187i$
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  + (−0.995 + 0.0950i)2-s + (0.396 − 0.254i)3-s + (0.981 − 0.189i)4-s + (−0.370 + 0.291i)6-s + (−0.959 + 0.281i)8-s + (−0.323 + 0.707i)9-s + (1.50 + 1.18i)11-s + (0.341 − 0.325i)12-s + (0.928 − 0.371i)16-s + (−0.642 − 0.123i)17-s + (0.254 − 0.734i)18-s + (1.07 − 1.51i)19-s + (−1.61 − 1.03i)22-s + (−0.308 + 0.356i)24-s + (−0.959 − 0.281i)25-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)2-s + (0.396 − 0.254i)3-s + (0.981 − 0.189i)4-s + (−0.370 + 0.291i)6-s + (−0.959 + 0.281i)8-s + (−0.323 + 0.707i)9-s + (1.50 + 1.18i)11-s + (0.341 − 0.325i)12-s + (0.928 − 0.371i)16-s + (−0.642 − 0.123i)17-s + (0.254 − 0.734i)18-s + (1.07 − 1.51i)19-s + (−1.61 − 1.03i)22-s + (−0.308 + 0.356i)24-s + (−0.959 − 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.982 - 0.187i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (419, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.982 - 0.187i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6948736110\)
\(L(\frac12)\) \(\approx\) \(0.6948736110\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.995 - 0.0950i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (-0.396 + 0.254i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.327 - 0.945i)T^{2} \)
11 \( 1 + (-1.50 - 1.18i)T + (0.235 + 0.971i)T^{2} \)
13 \( 1 + (0.888 - 0.458i)T^{2} \)
17 \( 1 + (0.642 + 0.123i)T + (0.928 + 0.371i)T^{2} \)
19 \( 1 + (-1.07 + 1.51i)T + (-0.327 - 0.945i)T^{2} \)
23 \( 1 + (-0.580 - 0.814i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.888 + 0.458i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (0.271 + 0.785i)T + (-0.786 + 0.618i)T^{2} \)
43 \( 1 + (-1.02 + 1.18i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (0.995 - 0.0950i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (1.38 - 0.407i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (-0.235 + 0.971i)T^{2} \)
71 \( 1 + (-0.928 + 0.371i)T^{2} \)
73 \( 1 + (1.54 - 1.21i)T + (0.235 - 0.971i)T^{2} \)
79 \( 1 + (-0.0475 + 0.998i)T^{2} \)
83 \( 1 + (0.264 - 0.105i)T + (0.723 - 0.690i)T^{2} \)
89 \( 1 + (1.67 + 1.07i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + (0.981 + 1.70i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.07080051734138537717900017158, −9.933834800421357435750268852147, −9.202864280248118305688536910518, −8.601581825872844664329934819142, −7.31439477935093135093062017042, −7.06854714196436060699481830370, −5.76027820241299323055786046252, −4.39091554942232282356443813882, −2.76456282467137060877622230801, −1.67418817551595387719434100082, 1.40486506447869574628400611934, 3.15146009486399960560981690455, 3.87400967809046699814197577673, 5.92073980492392829922034946796, 6.43304232167198870424123805398, 7.73466062822486655281106521340, 8.526244120843584957118020588682, 9.307128714684571693800387585777, 9.792060919711475165429666605232, 11.01875671426376306739607797861

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