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Properties

Label 2-513-171.122-c1-0-3
Degree $2$
Conductor $513$
Sign $0.929 - 0.369i$
Analytic cond. $4.09632$
Root an. cond. $2.02393$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.83·2-s + 1.37·4-s + (1.22 − 0.706i)5-s + (0.660 + 1.14i)7-s + 1.15·8-s + (−2.24 + 1.29i)10-s + (0.206 − 0.119i)11-s + 3.50i·13-s + (−1.21 − 2.09i)14-s − 4.86·16-s + (2.07 + 1.19i)17-s + (0.782 − 4.28i)19-s + (1.67 − 0.969i)20-s + (−0.380 + 0.219i)22-s − 3.19i·23-s + ⋯
L(s)  = 1  − 1.29·2-s + 0.686·4-s + (0.546 − 0.315i)5-s + (0.249 + 0.432i)7-s + 0.407·8-s + (−0.710 + 0.410i)10-s + (0.0623 − 0.0360i)11-s + 0.973i·13-s + (−0.324 − 0.561i)14-s − 1.21·16-s + (0.503 + 0.290i)17-s + (0.179 − 0.983i)19-s + (0.375 − 0.216i)20-s + (−0.0810 + 0.0467i)22-s − 0.665i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.369i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.929 - 0.369i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.929 - 0.369i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.804123 + 0.154171i\)
\(L(\frac12)\) \(\approx\) \(0.804123 + 0.154171i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.782 + 4.28i)T \)
good2 \( 1 + 1.83T + 2T^{2} \)
5 \( 1 + (-1.22 + 0.706i)T + (2.5 - 4.33i)T^{2} \)
7 \( 1 + (-0.660 - 1.14i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.206 + 0.119i)T + (5.5 - 9.52i)T^{2} \)
13 \( 1 - 3.50iT - 13T^{2} \)
17 \( 1 + (-2.07 - 1.19i)T + (8.5 + 14.7i)T^{2} \)
23 \( 1 + 3.19iT - 23T^{2} \)
29 \( 1 + (1.89 - 3.28i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-8.02 - 4.63i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + 1.94iT - 37T^{2} \)
41 \( 1 + (-0.360 - 0.624i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 - 6.41T + 43T^{2} \)
47 \( 1 + (-8.12 - 4.69i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.726 - 1.25i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-4.44 - 7.70i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.98 + 3.44i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 + (1.40 - 2.42i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-6.08 + 10.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 - 5.99iT - 79T^{2} \)
83 \( 1 + (-10.4 + 6.02i)T + (41.5 - 71.8i)T^{2} \)
89 \( 1 + (0.686 + 1.18i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 5.92iT - 97T^{2} \)
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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

−10.71162751773980651873826793047, −9.855661267268272545136672625566, −9.044197115064048954186810354435, −8.648471283485588291936639227398, −7.52006005783940274041414516382, −6.63699670803751470421470837745, −5.40633878259428867693857863371, −4.32154008580787147832664090137, −2.43967178543460386365861801679, −1.21051336989863332774453995570, 0.938670932447748719526798462640, 2.38479968765833709938643676513, 3.97503344855238793380700332625, 5.39353675963430860689721130889, 6.44342890532925033758432735740, 7.73171221723834735798182852469, 7.961760506294332540018091448777, 9.229921016327157395690324386559, 10.01371298542583612103205259687, 10.40348142925517161439686744450

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