Properties

Label 2-513-171.94-c0-0-1
Degree $2$
Conductor $513$
Sign $0.642 + 0.766i$
Analytic cond. $0.256020$
Root an. cond. $0.505984$
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  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + i·8-s − 0.999i·10-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.866 − 0.499i)14-s + (0.5 + 0.866i)16-s + i·19-s + (−0.866 − 0.499i)22-s + (−0.5 + 0.866i)23-s + 0.999·26-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)2-s + (0.5 − 0.866i)5-s + (−0.5 − 0.866i)7-s + i·8-s − 0.999i·10-s + (−0.5 − 0.866i)11-s + (0.866 + 0.5i)13-s + (−0.866 − 0.499i)14-s + (0.5 + 0.866i)16-s + i·19-s + (−0.866 − 0.499i)22-s + (−0.5 + 0.866i)23-s + 0.999·26-s + (−0.866 + 0.5i)29-s + (−0.866 − 0.5i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.642 + 0.766i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.642 + 0.766i$
Analytic conductor: \(0.256020\)
Root analytic conductor: \(0.505984\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :0),\ 0.642 + 0.766i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.299870122\)
\(L(\frac12)\) \(\approx\) \(1.299870122\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
19 \( 1 - iT \)
good2 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
13 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
23 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
43 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (-0.866 - 0.5i)T + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (0.866 + 0.5i)T + (0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + (-0.866 + 0.5i)T + (0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
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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

−11.11608376903383947611073060869, −10.31476321925093163428323810337, −9.175827320676644931459377038857, −8.436999809863312152515693868383, −7.39396094766092065689559899897, −5.92463756420725268380063423592, −5.33159935142862992574173459029, −4.00967768006003595161956318053, −3.43114659545402965708944077461, −1.70698425467015301675357382800, 2.35613662304357070568339692554, 3.47937554182328608214627523073, 4.81596675002654145887652361380, 5.74280609945688652991791074215, 6.44200835497182320907356911083, 7.19288494309097930733188520048, 8.563618229165264025512380990804, 9.641284651254724291617551212586, 10.25100520079833272722434362181, 11.20378679104639951017662690216

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