Properties

Label 2-507-13.12-c3-0-48
Degree $2$
Conductor $507$
Sign $-0.999 - 0.0304i$
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  − 5.52i·2-s + 3·3-s − 22.4·4-s + 6.08i·5-s − 16.5i·6-s + 20.2i·7-s + 80.0i·8-s + 9·9-s + 33.5·10-s − 48.8i·11-s − 67.4·12-s + 111.·14-s + 18.2i·15-s + 262.·16-s + 37.7·17-s − 49.7i·18-s + ⋯
L(s)  = 1  − 1.95i·2-s + 0.577·3-s − 2.81·4-s + 0.543i·5-s − 1.12i·6-s + 1.09i·7-s + 3.53i·8-s + 0.333·9-s + 1.06·10-s − 1.33i·11-s − 1.62·12-s + 2.13·14-s + 0.314i·15-s + 4.09·16-s + 0.538·17-s − 0.650i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-0.999 - 0.0304i$
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: \((2,\ 507,\ (\ :3/2),\ -0.999 - 0.0304i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.591672016\)
\(L(\frac12)\) \(\approx\) \(1.591672016\)
\(L(\frac{5}{2})\) 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 \( 1 + 5.52iT - 8T^{2} \)
5 \( 1 - 6.08iT - 125T^{2} \)
7 \( 1 - 20.2iT - 343T^{2} \)
11 \( 1 + 48.8iT - 1.33e3T^{2} \)
17 \( 1 - 37.7T + 4.91e3T^{2} \)
19 \( 1 + 120. iT - 6.85e3T^{2} \)
23 \( 1 + 74.8T + 1.21e4T^{2} \)
29 \( 1 + 112.T + 2.43e4T^{2} \)
31 \( 1 + 113. iT - 2.97e4T^{2} \)
37 \( 1 + 85.7iT - 5.06e4T^{2} \)
41 \( 1 - 133. iT - 6.89e4T^{2} \)
43 \( 1 - 319.T + 7.95e4T^{2} \)
47 \( 1 + 401. iT - 1.03e5T^{2} \)
53 \( 1 + 384.T + 1.48e5T^{2} \)
59 \( 1 + 121. iT - 2.05e5T^{2} \)
61 \( 1 - 220.T + 2.26e5T^{2} \)
67 \( 1 + 975. iT - 3.00e5T^{2} \)
71 \( 1 + 106. iT - 3.57e5T^{2} \)
73 \( 1 - 43.2iT - 3.89e5T^{2} \)
79 \( 1 - 539.T + 4.93e5T^{2} \)
83 \( 1 + 811. iT - 5.71e5T^{2} \)
89 \( 1 + 1.13e3iT - 7.04e5T^{2} \)
97 \( 1 - 229. iT - 9.12e5T^{2} \)
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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

−10.21400459978529186402667748694, −9.183499043334735361433286721467, −8.835276471109914030932702389049, −7.83410811378893732382319964721, −5.99293141299598908635233508596, −4.93437384518902995645841634622, −3.60431005044099259703543554178, −2.89408455034463980604846085952, −2.10088191627051048717347409731, −0.53773409931947617708394179075, 1.19059553782895418809384323532, 3.80214708872674321832120147251, 4.44535144714234415771825996676, 5.44402159536853095745613949676, 6.59224781714392764679765697423, 7.50216274094252138004732867631, 7.86233768450993982921652377687, 8.883796811634641569852833856946, 9.746781724253678932882056895062, 10.31757376672630082504318158506

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