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Properties

Label 2-507-13.12-c3-0-18
Degree $2$
Conductor $507$
Sign $-0.691 + 0.722i$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.23i·2-s − 3·3-s − 2.47·4-s + 13.5i·5-s − 9.70i·6-s + 1.42i·7-s + 17.8i·8-s + 9·9-s − 43.9·10-s + 54.5i·11-s + 7.42·12-s − 4.62·14-s − 40.7i·15-s − 77.6·16-s + 114.·17-s + 29.1i·18-s + ⋯
L(s)  = 1  + 1.14i·2-s − 0.577·3-s − 0.309·4-s + 1.21i·5-s − 0.660i·6-s + 0.0771i·7-s + 0.790i·8-s + 0.333·9-s − 1.39·10-s + 1.49i·11-s + 0.178·12-s − 0.0883·14-s − 0.701i·15-s − 1.21·16-s + 1.63·17-s + 0.381i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.691 + 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.691 + 0.722i$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (337, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -0.691 + 0.722i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.607585559\)
\(L(\frac12)\) \(\approx\) \(1.607585559\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + 3T \)
13 \( 1 \)
good2 \( 1 - 3.23iT - 8T^{2} \)
5 \( 1 - 13.5iT - 125T^{2} \)
7 \( 1 - 1.42iT - 343T^{2} \)
11 \( 1 - 54.5iT - 1.33e3T^{2} \)
17 \( 1 - 114.T + 4.91e3T^{2} \)
19 \( 1 - 104. iT - 6.85e3T^{2} \)
23 \( 1 - 64.5T + 1.21e4T^{2} \)
29 \( 1 + 60.8T + 2.43e4T^{2} \)
31 \( 1 - 148. iT - 2.97e4T^{2} \)
37 \( 1 + 20.9iT - 5.06e4T^{2} \)
41 \( 1 + 371. iT - 6.89e4T^{2} \)
43 \( 1 + 40.2T + 7.95e4T^{2} \)
47 \( 1 - 639. iT - 1.03e5T^{2} \)
53 \( 1 - 102.T + 1.48e5T^{2} \)
59 \( 1 + 704. iT - 2.05e5T^{2} \)
61 \( 1 + 819.T + 2.26e5T^{2} \)
67 \( 1 + 574. iT - 3.00e5T^{2} \)
71 \( 1 + 365. iT - 3.57e5T^{2} \)
73 \( 1 + 965. iT - 3.89e5T^{2} \)
79 \( 1 - 580.T + 4.93e5T^{2} \)
83 \( 1 + 175. iT - 5.71e5T^{2} \)
89 \( 1 + 20.0iT - 7.04e5T^{2} \)
97 \( 1 + 1.22e3iT - 9.12e5T^{2} \)
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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

−10.83879430230995189875280084390, −10.30410764860260107333892210270, −9.291239467442467662838659136612, −7.74031546972296114518575223224, −7.41307990416009435987001293124, −6.52543306931362447020578572321, −5.76266207262730564870187021403, −4.80318532788939101928736191679, −3.30747590455841235709702811688, −1.84779260447903593185903640195, 0.61538407868374294310533134902, 1.17098645101626331140246385563, 2.85266841437859537536953970397, 3.94033872462662468449802746938, 5.09741257850625323624268607470, 5.93889960031058456978434427473, 7.21196116895121582827964104468, 8.441798041556964190797495972898, 9.250336850590661752565736796427, 10.12920371943718620128035860026

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