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Properties

Label 2-536-536.91-c0-0-0
Degree $2$
Conductor $536$
Sign $-0.195 + 0.980i$
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  + (0.415 − 0.909i)2-s + (−1.61 + 1.03i)3-s + (−0.654 − 0.755i)4-s + (0.273 + 1.89i)6-s + (−0.959 + 0.281i)8-s + (1.11 − 2.44i)9-s + (0.273 − 1.89i)11-s + (1.84 + 0.540i)12-s + (−0.142 + 0.989i)16-s + (0.857 − 0.989i)17-s + (−1.75 − 2.02i)18-s + (−0.118 − 0.258i)19-s + (−1.61 − 1.03i)22-s + (1.25 − 1.45i)24-s + (−0.959 − 0.281i)25-s + ⋯
L(s)  = 1  + (0.415 − 0.909i)2-s + (−1.61 + 1.03i)3-s + (−0.654 − 0.755i)4-s + (0.273 + 1.89i)6-s + (−0.959 + 0.281i)8-s + (1.11 − 2.44i)9-s + (0.273 − 1.89i)11-s + (1.84 + 0.540i)12-s + (−0.142 + 0.989i)16-s + (0.857 − 0.989i)17-s + (−1.75 − 2.02i)18-s + (−0.118 − 0.258i)19-s + (−1.61 − 1.03i)22-s + (1.25 − 1.45i)24-s + (−0.959 − 0.281i)25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $-0.195 + 0.980i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (91, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ -0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5593964322\)
\(L(\frac12)\) \(\approx\) \(0.5593964322\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.415 + 0.909i)T \)
67 \( 1 + (0.142 - 0.989i)T \)
good3 \( 1 + (1.61 - 1.03i)T + (0.415 - 0.909i)T^{2} \)
5 \( 1 + (0.959 + 0.281i)T^{2} \)
7 \( 1 + (0.654 + 0.755i)T^{2} \)
11 \( 1 + (-0.273 + 1.89i)T + (-0.959 - 0.281i)T^{2} \)
13 \( 1 + (-0.841 - 0.540i)T^{2} \)
17 \( 1 + (-0.857 + 0.989i)T + (-0.142 - 0.989i)T^{2} \)
19 \( 1 + (0.118 + 0.258i)T + (-0.654 + 0.755i)T^{2} \)
23 \( 1 + (-0.415 + 0.909i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (-0.841 + 0.540i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.544 - 0.627i)T + (-0.142 - 0.989i)T^{2} \)
43 \( 1 + (-0.186 + 0.215i)T + (-0.142 - 0.989i)T^{2} \)
47 \( 1 + (-0.415 + 0.909i)T^{2} \)
53 \( 1 + (0.142 - 0.989i)T^{2} \)
59 \( 1 + (-1.84 + 0.540i)T + (0.841 - 0.540i)T^{2} \)
61 \( 1 + (0.959 - 0.281i)T^{2} \)
71 \( 1 + (0.142 - 0.989i)T^{2} \)
73 \( 1 + (-0.186 - 1.29i)T + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (-0.841 - 0.540i)T^{2} \)
83 \( 1 + (-0.0405 + 0.281i)T + (-0.959 - 0.281i)T^{2} \)
89 \( 1 + (-0.698 - 0.449i)T + (0.415 + 0.909i)T^{2} \)
97 \( 1 + 1.30T + T^{2} \)
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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

−11.05424758337473666094261948563, −10.12982816034008004955066822096, −9.563097824275529806908025764554, −8.516106610227950084063113151438, −6.62084264253245422019102599768, −5.72035793527572216764793772269, −5.23177179783834299965665814644, −4.07272112170954585319457100834, −3.23969147257905718551291939373, −0.77503699253638138875629990824, 1.80226774724168168938452913565, 4.12779192964502698574452147278, 5.07741464139655434511637793477, 5.88447119781656295968453292885, 6.68243142199702140249961621299, 7.40380732394430415798655966428, 8.038630650625263907556076904645, 9.637961892356594203224925086211, 10.47906125963519112075945652640, 11.72776642581654457822196870283

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