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Properties

Label 2-810-135.122-c1-0-0
Degree $2$
Conductor $810$
Sign $0.864 - 0.501i$
Analytic cond. $6.46788$
Root an. cond. $2.54320$
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  + (−0.819 − 0.573i)2-s + (0.342 + 0.939i)4-s + (−1.82 − 1.29i)5-s + (0.960 + 2.06i)7-s + (0.258 − 0.965i)8-s + (0.749 + 2.10i)10-s + (0.117 + 0.139i)11-s + (−0.983 − 1.40i)13-s + (0.394 − 2.23i)14-s + (−0.766 + 0.642i)16-s + (0.952 + 3.55i)17-s + (0.0524 − 0.0303i)19-s + (0.594 − 2.15i)20-s + (−0.0159 − 0.182i)22-s + (−0.314 − 0.146i)23-s + ⋯
L(s)  = 1  + (−0.579 − 0.405i)2-s + (0.171 + 0.469i)4-s + (−0.814 − 0.579i)5-s + (0.363 + 0.778i)7-s + (0.0915 − 0.341i)8-s + (0.236 + 0.666i)10-s + (0.0354 + 0.0421i)11-s + (−0.272 − 0.389i)13-s + (0.105 − 0.598i)14-s + (−0.191 + 0.160i)16-s + (0.231 + 0.862i)17-s + (0.0120 − 0.00695i)19-s + (0.133 − 0.481i)20-s + (−0.00339 − 0.0388i)22-s + (−0.0655 − 0.0305i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.864 - 0.501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 810 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.864 - 0.501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(810\)    =    \(2 \cdot 3^{4} \cdot 5\)
Sign: $0.864 - 0.501i$
Analytic conductor: \(6.46788\)
Root analytic conductor: \(2.54320\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{810} (557, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 810,\ (\ :1/2),\ 0.864 - 0.501i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.865085 + 0.232806i\)
\(L(\frac12)\) \(\approx\) \(0.865085 + 0.232806i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.819 + 0.573i)T \)
3 \( 1 \)
5 \( 1 + (1.82 + 1.29i)T \)
good7 \( 1 + (-0.960 - 2.06i)T + (-4.49 + 5.36i)T^{2} \)
11 \( 1 + (-0.117 - 0.139i)T + (-1.91 + 10.8i)T^{2} \)
13 \( 1 + (0.983 + 1.40i)T + (-4.44 + 12.2i)T^{2} \)
17 \( 1 + (-0.952 - 3.55i)T + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.0524 + 0.0303i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.314 + 0.146i)T + (14.7 + 17.6i)T^{2} \)
29 \( 1 + (-1.26 - 7.17i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (-2.53 + 0.924i)T + (23.7 - 19.9i)T^{2} \)
37 \( 1 + (3.48 - 0.934i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-11.0 - 1.95i)T + (38.5 + 14.0i)T^{2} \)
43 \( 1 + (0.788 - 9.00i)T + (-42.3 - 7.46i)T^{2} \)
47 \( 1 + (-5.68 + 2.64i)T + (30.2 - 36.0i)T^{2} \)
53 \( 1 + (-6.90 - 6.90i)T + 53iT^{2} \)
59 \( 1 + (8.74 + 7.34i)T + (10.2 + 58.1i)T^{2} \)
61 \( 1 + (-12.0 - 4.37i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (10.2 - 7.17i)T + (22.9 - 62.9i)T^{2} \)
71 \( 1 + (-7.52 - 4.34i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (5.44 + 1.45i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-15.5 + 2.73i)T + (74.2 - 27.0i)T^{2} \)
83 \( 1 + (7.83 - 11.1i)T + (-28.3 - 77.9i)T^{2} \)
89 \( 1 + (5.06 + 8.77i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-6.26 - 0.548i)T + (95.5 + 16.8i)T^{2} \)
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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

−10.38097950381109798919041781882, −9.328017771275030590973300747902, −8.613118372041508073922516512600, −8.028808058698400239780650306617, −7.17456078426038533304332265690, −5.87445199923077064594161382246, −4.86209786491218815147488073234, −3.81381803484568018403838596650, −2.62888363973636836885364181716, −1.19841448481696628254049151741, 0.63467225874477216189698470452, 2.44259359878422190015504732202, 3.83621068399559322147048431314, 4.72536721131474666285564041868, 6.00681355242445351318079281551, 7.09825128215874101991399654893, 7.47143506997646022407639696553, 8.321371547692752483666650428910, 9.306784253822625411849176181717, 10.22094439137342878446454927837

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