Properties

Label 2-536-536.211-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.0864 - 0.996i$
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  + (0.723 + 0.690i)2-s + (0.0930 + 0.647i)3-s + (0.0475 + 0.998i)4-s + (−0.379 + 0.532i)6-s + (−0.654 + 0.755i)8-s + (0.548 − 0.161i)9-s + (−0.759 − 1.06i)11-s + (−0.642 + 0.123i)12-s + (−0.995 + 0.0950i)16-s + (−0.0845 + 1.77i)17-s + (0.508 + 0.262i)18-s + (−0.469 − 1.93i)19-s + (0.186 − 1.29i)22-s + (−0.550 − 0.353i)24-s + (−0.654 − 0.755i)25-s + ⋯
L(s)  = 1  + (0.723 + 0.690i)2-s + (0.0930 + 0.647i)3-s + (0.0475 + 0.998i)4-s + (−0.379 + 0.532i)6-s + (−0.654 + 0.755i)8-s + (0.548 − 0.161i)9-s + (−0.759 − 1.06i)11-s + (−0.642 + 0.123i)12-s + (−0.995 + 0.0950i)16-s + (−0.0845 + 1.77i)17-s + (0.508 + 0.262i)18-s + (−0.469 − 1.93i)19-s + (0.186 − 1.29i)22-s + (−0.550 − 0.353i)24-s + (−0.654 − 0.755i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0864 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.0864 - 0.996i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.0864 - 0.996i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.278722415\)
\(L(\frac12)\) \(\approx\) \(1.278722415\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.723 - 0.690i)T \)
67 \( 1 + (-0.415 + 0.909i)T \)
good3 \( 1 + (-0.0930 - 0.647i)T + (-0.959 + 0.281i)T^{2} \)
5 \( 1 + (0.654 + 0.755i)T^{2} \)
7 \( 1 + (0.888 + 0.458i)T^{2} \)
11 \( 1 + (0.759 + 1.06i)T + (-0.327 + 0.945i)T^{2} \)
13 \( 1 + (0.786 + 0.618i)T^{2} \)
17 \( 1 + (0.0845 - 1.77i)T + (-0.995 - 0.0950i)T^{2} \)
19 \( 1 + (0.469 + 1.93i)T + (-0.888 + 0.458i)T^{2} \)
23 \( 1 + (-0.235 + 0.971i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.786 - 0.618i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (-1.70 + 0.879i)T + (0.580 - 0.814i)T^{2} \)
43 \( 1 + (-0.975 - 0.627i)T + (0.415 + 0.909i)T^{2} \)
47 \( 1 + (-0.723 - 0.690i)T^{2} \)
53 \( 1 + (-0.415 + 0.909i)T^{2} \)
59 \( 1 + (1.28 - 1.48i)T + (-0.142 - 0.989i)T^{2} \)
61 \( 1 + (0.327 + 0.945i)T^{2} \)
71 \( 1 + (0.995 - 0.0950i)T^{2} \)
73 \( 1 + (-0.0552 + 0.0775i)T + (-0.327 - 0.945i)T^{2} \)
79 \( 1 + (-0.928 + 0.371i)T^{2} \)
83 \( 1 + (0.827 - 0.0789i)T + (0.981 - 0.189i)T^{2} \)
89 \( 1 + (0.205 - 1.43i)T + (-0.959 - 0.281i)T^{2} \)
97 \( 1 + (0.0475 + 0.0824i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.08621978435671663569381997418, −10.64519210921574197212182939896, −9.309137987663014876376722320508, −8.525089563652020350103469186846, −7.67041950023874865248879903800, −6.53062422800262724986476219596, −5.73760427596897279897482345193, −4.59911520141905224933369458715, −3.86307405577615865936142914033, −2.65059047860534385449777820188, 1.64022771488741472356352291560, 2.66592575198728962847384038119, 4.10755703309075446505343039832, 5.02865963947302203505757943598, 6.08806245356202248290286319653, 7.21806750917104312853787945435, 7.84505322249012947733096316904, 9.465613368686473030114008830684, 9.974424988901836271190250034875, 10.92798869869932711337707468526

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