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Properties

Label 2-536-536.171-c0-0-0
Degree $2$
Conductor $536$
Sign $0.809 - 0.586i$
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  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + (−1 + 1.73i)11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 1.99·22-s − 0.999·24-s + 25-s + 27-s + (−0.499 + 0.866i)32-s + (1 − 1.73i)33-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (0.5 + 0.866i)6-s + 0.999·8-s + (−1 + 1.73i)11-s + (0.499 − 0.866i)12-s + (−0.5 − 0.866i)16-s + (0.5 + 0.866i)17-s + (0.5 + 0.866i)19-s + 1.99·22-s − 0.999·24-s + 25-s + 27-s + (−0.499 + 0.866i)32-s + (1 − 1.73i)33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.809 - 0.586i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(536\)    =    \(2^{3} \cdot 67\)
Sign: $0.809 - 0.586i$
Analytic conductor: \(0.267498\)
Root analytic conductor: \(0.517202\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{536} (171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 536,\ (\ :0),\ 0.809 - 0.586i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3803774120\)
\(L(\frac12)\) \(\approx\) \(0.3803774120\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 - T \)
good3 \( 1 + T + T^{2} \)
5 \( 1 - T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + (0.5 - 0.866i)T^{2} \)
41 \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \)
43 \( 1 + T + T^{2} \)
47 \( 1 + (0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + T + T^{2} \)
61 \( 1 + (0.5 - 0.866i)T^{2} \)
71 \( 1 + (0.5 + 0.866i)T^{2} \)
73 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (1 + 1.73i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.10335658118808031161830648786, −10.13706513807319312658730735032, −9.938627514309825610126949123469, −8.497049087510355042284163808275, −7.70079764087139953990911230924, −6.66934095096703410503043596149, −5.31081741553311505993037638099, −4.58318806112309793278024837611, −3.12938971806492192347521808929, −1.68280643249630558567291777965, 0.63829936192861985863881485190, 3.07810585574063990521912206915, 4.99542170052193975795754294078, 5.43002191694531996187168390264, 6.33419863455215072462065012761, 7.25067456461165671880691123857, 8.297753221193340614514320284687, 8.991013824734288473957996174058, 10.15304715437034814731109041774, 10.95284832969294740838734661737

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