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Properties

Label 2-513-171.7-c1-0-4
Degree $2$
Conductor $513$
Sign $0.210 - 0.977i$
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  − 0.539·2-s − 1.70·4-s + (−0.473 + 0.820i)5-s + (1.18 − 2.05i)7-s + 1.99·8-s + (0.255 − 0.442i)10-s + (−1.76 + 3.05i)11-s − 1.02·13-s + (−0.638 + 1.10i)14-s + 2.34·16-s + (0.347 + 0.602i)17-s + (2.46 + 3.59i)19-s + (0.809 − 1.40i)20-s + (0.951 − 1.64i)22-s − 3.38·23-s + ⋯
L(s)  = 1  − 0.381·2-s − 0.854·4-s + (−0.211 + 0.366i)5-s + (0.447 − 0.775i)7-s + 0.706·8-s + (0.0807 − 0.139i)10-s + (−0.532 + 0.922i)11-s − 0.285·13-s + (−0.170 + 0.295i)14-s + 0.585·16-s + (0.0843 + 0.146i)17-s + (0.565 + 0.824i)19-s + (0.181 − 0.313i)20-s + (0.202 − 0.351i)22-s − 0.704·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 513 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.210 - 0.977i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(513\)    =    \(3^{3} \cdot 19\)
Sign: $0.210 - 0.977i$
Analytic conductor: \(4.09632\)
Root analytic conductor: \(2.02393\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{513} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 513,\ (\ :1/2),\ 0.210 - 0.977i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.597189 + 0.482129i\)
\(L(\frac12)\) \(\approx\) \(0.597189 + 0.482129i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.46 - 3.59i)T \)
good2 \( 1 + 0.539T + 2T^{2} \)
5 \( 1 + (0.473 - 0.820i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + (-1.18 + 2.05i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.76 - 3.05i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + 1.02T + 13T^{2} \)
17 \( 1 + (-0.347 - 0.602i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + 3.38T + 23T^{2} \)
29 \( 1 + (-1.76 - 3.05i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.48 - 7.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 0.345T + 37T^{2} \)
41 \( 1 + (5.69 - 9.85i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + 4.21T + 43T^{2} \)
47 \( 1 + (-5.12 - 8.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-3.33 + 5.77i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.53 + 9.59i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.73 + 3.00i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + 2.04T + 67T^{2} \)
71 \( 1 + (1.75 + 3.04i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.57 + 7.92i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 - 13.7T + 79T^{2} \)
83 \( 1 + (2.41 - 4.18i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-0.902 + 1.56i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.0T + 97T^{2} \)
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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

−10.74688821521117085704230948170, −10.18725347679408342624188295639, −9.472560392156917748927528741712, −8.226975022312324401065329365814, −7.70256154408108079409193861106, −6.77173572798874398487212197151, −5.20539166449153753293393491376, −4.48650179385041103276763245677, −3.32075449533082448942097580398, −1.43179135473576075537143844019, 0.58350803118510297903593002759, 2.51257206467317802501417643662, 4.04433587920248388622435721952, 5.07593346553737583135665913550, 5.80313334994830970823592603221, 7.35158152650981437565055662293, 8.407353285196629614430459888754, 8.655995166666298793075040363189, 9.737987748492338764045735809366, 10.53004587541997954801732804036

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