Properties

Label 2-507-1.1-c3-0-49
Degree $2$
Conductor $507$
Sign $-1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s − 8·4-s + 5.19·5-s + 10.3·7-s + 9·9-s − 51.9·11-s + 24·12-s − 15.5·15-s + 64·16-s + 117·17-s + 24.2·19-s − 41.5·20-s − 31.1·21-s − 18·23-s − 98·25-s − 27·27-s − 83.1·28-s − 99·29-s + 193.·31-s + 155.·33-s + 54·35-s − 72·36-s − 112.·37-s − 36.3·41-s + 82·43-s + 415.·44-s + 46.7·45-s + ⋯
L(s)  = 1  − 0.577·3-s − 4-s + 0.464·5-s + 0.561·7-s + 0.333·9-s − 1.42·11-s + 0.577·12-s − 0.268·15-s + 16-s + 1.66·17-s + 0.292·19-s − 0.464·20-s − 0.323·21-s − 0.163·23-s − 0.784·25-s − 0.192·27-s − 0.561·28-s − 0.633·29-s + 1.12·31-s + 0.822·33-s + 0.260·35-s − 0.333·36-s − 0.500·37-s − 0.138·41-s + 0.290·43-s + 1.42·44-s + 0.154·45-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) 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 \( 1 + 8T^{2} \)
5 \( 1 - 5.19T + 125T^{2} \)
7 \( 1 - 10.3T + 343T^{2} \)
11 \( 1 + 51.9T + 1.33e3T^{2} \)
17 \( 1 - 117T + 4.91e3T^{2} \)
19 \( 1 - 24.2T + 6.85e3T^{2} \)
23 \( 1 + 18T + 1.21e4T^{2} \)
29 \( 1 + 99T + 2.43e4T^{2} \)
31 \( 1 - 193.T + 2.97e4T^{2} \)
37 \( 1 + 112.T + 5.06e4T^{2} \)
41 \( 1 + 36.3T + 6.89e4T^{2} \)
43 \( 1 - 82T + 7.95e4T^{2} \)
47 \( 1 - 72.7T + 1.03e5T^{2} \)
53 \( 1 + 261T + 1.48e5T^{2} \)
59 \( 1 + 789.T + 2.05e5T^{2} \)
61 \( 1 + 719T + 2.26e5T^{2} \)
67 \( 1 + 703.T + 3.00e5T^{2} \)
71 \( 1 - 467.T + 3.57e5T^{2} \)
73 \( 1 + 684.T + 3.89e5T^{2} \)
79 \( 1 + 440T + 4.93e5T^{2} \)
83 \( 1 - 1.19e3T + 5.71e5T^{2} \)
89 \( 1 + 1.51e3T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3T + 9.12e5T^{2} \)
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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

−10.08862902021295803199623194871, −9.364871664537967415666451677738, −8.065609322373533780316401465237, −7.63909149634327756182825372380, −5.94143941888053716067570398777, −5.34814836287692396252486149375, −4.54148079788924452929699143962, −3.13774386471494526854015675471, −1.42810403971386508661531117957, 0, 1.42810403971386508661531117957, 3.13774386471494526854015675471, 4.54148079788924452929699143962, 5.34814836287692396252486149375, 5.94143941888053716067570398777, 7.63909149634327756182825372380, 8.065609322373533780316401465237, 9.364871664537967415666451677738, 10.08862902021295803199623194871

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