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Properties

Label 2-507-13.12-c3-0-31
Degree $2$
Conductor $507$
Sign $0.832 + 0.554i$
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  − 2.74i·2-s − 3·3-s + 0.483·4-s + 19.4i·5-s + 8.22i·6-s − 7.48i·7-s − 23.2i·8-s + 9·9-s + 53.4·10-s − 22.8i·11-s − 1.44·12-s − 20.5·14-s − 58.4i·15-s − 59.8·16-s − 67.0·17-s − 24.6i·18-s + ⋯
L(s)  = 1  − 0.969i·2-s − 0.577·3-s + 0.0604·4-s + 1.74i·5-s + 0.559i·6-s − 0.404i·7-s − 1.02i·8-s + 0.333·9-s + 1.68·10-s − 0.627i·11-s − 0.0348·12-s − 0.391·14-s − 1.00i·15-s − 0.935·16-s − 0.956·17-s − 0.323i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.832 + 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.832 + 0.554i$
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: \((2,\ 507,\ (\ :3/2),\ 0.832 + 0.554i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.770981843\)
\(L(\frac12)\) \(\approx\) \(1.770981843\)
\(L(\frac{5}{2})\) 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 \( 1 + 2.74iT - 8T^{2} \)
5 \( 1 - 19.4iT - 125T^{2} \)
7 \( 1 + 7.48iT - 343T^{2} \)
11 \( 1 + 22.8iT - 1.33e3T^{2} \)
17 \( 1 + 67.0T + 4.91e3T^{2} \)
19 \( 1 - 16.5iT - 6.85e3T^{2} \)
23 \( 1 - 175.T + 1.21e4T^{2} \)
29 \( 1 - 291.T + 2.43e4T^{2} \)
31 \( 1 - 117. iT - 2.97e4T^{2} \)
37 \( 1 - 154. iT - 5.06e4T^{2} \)
41 \( 1 + 251. iT - 6.89e4T^{2} \)
43 \( 1 - 502.T + 7.95e4T^{2} \)
47 \( 1 - 281. iT - 1.03e5T^{2} \)
53 \( 1 - 366.T + 1.48e5T^{2} \)
59 \( 1 - 79.6iT - 2.05e5T^{2} \)
61 \( 1 + 194.T + 2.26e5T^{2} \)
67 \( 1 - 400. iT - 3.00e5T^{2} \)
71 \( 1 - 528. iT - 3.57e5T^{2} \)
73 \( 1 - 734. iT - 3.89e5T^{2} \)
79 \( 1 - 113.T + 4.93e5T^{2} \)
83 \( 1 + 933. iT - 5.71e5T^{2} \)
89 \( 1 + 1.19e3iT - 7.04e5T^{2} \)
97 \( 1 - 557. iT - 9.12e5T^{2} \)
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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

−10.57834774807418439416381030137, −10.18916726648669030622836829095, −8.876707808263926212745749505627, −7.27932376684632553745099187338, −6.82203721092315154578798768964, −5.99195795285511548579597156553, −4.34228264817185089066479656010, −3.21309916750399194607674742038, −2.49709171264340259838724122066, −0.900751033412110130673814195232, 0.813360163900939930093007862216, 2.26112744619240585251341044605, 4.50315957749976646393967911516, 5.00485715527943421656345051998, 5.90441007840390295105528093078, 6.83995212712680040001806267682, 7.82300430608083267562002155731, 8.778551997775667829280183316194, 9.268807314155556960549554166800, 10.65169215011825354306823852679

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