Properties

Label 2-507-169.10-c1-0-7
Degree $2$
Conductor $507$
Sign $0.664 + 0.747i$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
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  + (−2.42 − 0.494i)2-s + (−0.987 + 0.160i)3-s + (3.79 + 1.61i)4-s + (0.0746 − 0.302i)5-s + (2.47 + 0.0996i)6-s + (−4.01 + 1.90i)7-s + (−4.32 − 2.98i)8-s + (0.948 − 0.316i)9-s + (−0.330 + 0.697i)10-s + (1.26 − 3.80i)11-s + (−4.00 − 0.986i)12-s + (−1.26 + 3.37i)13-s + (10.6 − 2.63i)14-s + (−0.0250 + 0.310i)15-s + (3.29 + 3.42i)16-s + (−0.744 − 1.56i)17-s + ⋯
L(s)  = 1  + (−1.71 − 0.350i)2-s + (−0.569 + 0.0926i)3-s + (1.89 + 0.808i)4-s + (0.0333 − 0.135i)5-s + (1.00 + 0.0406i)6-s + (−1.51 + 0.719i)7-s + (−1.52 − 1.05i)8-s + (0.316 − 0.105i)9-s + (−0.104 + 0.220i)10-s + (0.382 − 1.14i)11-s + (−1.15 − 0.284i)12-s + (−0.352 + 0.935i)13-s + (2.85 − 0.703i)14-s + (−0.00648 + 0.0802i)15-s + (0.823 + 0.857i)16-s + (−0.180 − 0.380i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.664 + 0.747i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.664 + 0.747i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.664 + 0.747i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.318900 - 0.143143i\)
\(L(\frac12)\) \(\approx\) \(0.318900 - 0.143143i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.987 - 0.160i)T \)
13 \( 1 + (1.26 - 3.37i)T \)
good2 \( 1 + (2.42 + 0.494i)T + (1.83 + 0.783i)T^{2} \)
5 \( 1 + (-0.0746 + 0.302i)T + (-4.42 - 2.32i)T^{2} \)
7 \( 1 + (4.01 - 1.90i)T + (4.42 - 5.42i)T^{2} \)
11 \( 1 + (-1.26 + 3.80i)T + (-8.79 - 6.60i)T^{2} \)
17 \( 1 + (0.744 + 1.56i)T + (-10.7 + 13.1i)T^{2} \)
19 \( 1 + (2.40 + 1.39i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.703 - 1.21i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.978 - 4.79i)T + (-26.6 - 11.3i)T^{2} \)
31 \( 1 + (-2.45 - 4.67i)T + (-17.6 + 25.5i)T^{2} \)
37 \( 1 + (-5.54 + 8.76i)T + (-15.8 - 33.4i)T^{2} \)
41 \( 1 + (1.80 + 11.0i)T + (-38.8 + 12.9i)T^{2} \)
43 \( 1 + (-5.54 + 3.50i)T + (18.4 - 38.8i)T^{2} \)
47 \( 1 + (3.09 + 0.376i)T + (45.6 + 11.2i)T^{2} \)
53 \( 1 + (-5.82 + 8.43i)T + (-18.7 - 49.5i)T^{2} \)
59 \( 1 + (-1.93 - 1.86i)T + (2.37 + 58.9i)T^{2} \)
61 \( 1 + (-12.1 + 0.983i)T + (60.2 - 9.78i)T^{2} \)
67 \( 1 + (3.15 + 7.39i)T + (-46.4 + 48.3i)T^{2} \)
71 \( 1 + (-9.40 + 7.67i)T + (14.2 - 69.5i)T^{2} \)
73 \( 1 + (-8.26 - 9.32i)T + (-8.79 + 72.4i)T^{2} \)
79 \( 1 + (0.327 - 2.69i)T + (-76.7 - 18.9i)T^{2} \)
83 \( 1 + (6.01 - 2.28i)T + (62.1 - 55.0i)T^{2} \)
89 \( 1 + (2.48 - 1.43i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.220 - 0.0640i)T + (81.9 + 51.8i)T^{2} \)
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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

−10.71534540162296307637281076441, −9.708002030948777324121142134397, −9.065932665129385567948437656494, −8.652842705840264054869004690124, −7.03670424267093168140451173052, −6.63453269995762603586075697316, −5.47140472834414554540487684057, −3.55259752944211803066543761498, −2.33798009242984023481845490105, −0.55819140477721676234801524690, 0.854911077813306972666481716403, 2.61529649144123564049480629346, 4.34333613506509304864215913495, 6.15783673224432611423363722054, 6.60402166010181267032020165879, 7.43390002925110366271622313554, 8.288504674639742361273565378243, 9.637618610318093498981490757324, 9.921586165182003422112655649717, 10.54024578282525313962369396196

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