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Properties

Label 2-507-13.10-c1-0-22
Degree $2$
Conductor $507$
Sign $0.114 + 0.993i$
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  + (1.77 − 1.02i)2-s + (0.5 + 0.866i)3-s + (1.09 − 1.90i)4-s − 3.35i·5-s + (1.77 + 1.02i)6-s + (−1.94 − 1.12i)7-s − 0.405i·8-s + (−0.499 + 0.866i)9-s + (−3.43 − 5.95i)10-s + (4.27 − 2.46i)11-s + 2.19·12-s − 4.60·14-s + (2.90 − 1.67i)15-s + (1.78 + 3.08i)16-s + (0.455 − 0.789i)17-s + 2.04i·18-s + ⋯
L(s)  = 1  + (1.25 − 0.724i)2-s + (0.288 + 0.499i)3-s + (0.549 − 0.951i)4-s − 1.50i·5-s + (0.724 + 0.418i)6-s + (−0.735 − 0.424i)7-s − 0.143i·8-s + (−0.166 + 0.288i)9-s + (−1.08 − 1.88i)10-s + (1.28 − 0.744i)11-s + 0.634·12-s − 1.23·14-s + (0.750 − 0.433i)15-s + (0.445 + 0.771i)16-s + (0.110 − 0.191i)17-s + 0.482i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.114 + 0.993i$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.12611 - 1.89441i\)
\(L(\frac12)\) \(\approx\) \(2.12611 - 1.89441i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 \)
good2 \( 1 + (-1.77 + 1.02i)T + (1 - 1.73i)T^{2} \)
5 \( 1 + 3.35iT - 5T^{2} \)
7 \( 1 + (1.94 + 1.12i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (-4.27 + 2.46i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + (-0.455 + 0.789i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3.29 + 1.90i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.01 - 1.75i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.96 - 3.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 8.82iT - 31T^{2} \)
37 \( 1 + (-7.62 + 4.40i)T + (18.5 - 32.0i)T^{2} \)
41 \( 1 + (6.00 - 3.46i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (1.14 - 1.97i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 3.80iT - 47T^{2} \)
53 \( 1 - 0.542T + 53T^{2} \)
59 \( 1 + (-4.08 - 2.35i)T + (29.5 + 51.0i)T^{2} \)
61 \( 1 + (1.83 - 3.18i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.31 + 0.760i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (-2.05 - 1.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 - 7.41iT - 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 + 2.30iT - 83T^{2} \)
89 \( 1 + (8.71 - 5.02i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (13.9 + 8.06i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.98699776268778507898342544332, −9.880900882884335347992523311382, −8.962720183803426781810798675238, −8.403095324759946318279475142182, −6.69315058692235888439814166013, −5.60524366518547260374924464826, −4.67588579498723985676805113366, −3.97122200562950366872881727527, −3.06674744868005468948860916270, −1.30029249377825289363571086986, 2.36186860793356474826122172332, 3.43310146411995023878919577364, 4.27744407789312378467317151520, 5.93710724502780166732271676387, 6.50529980744778656586472388369, 6.95775983339680254543665070081, 7.995681696573254577736126700239, 9.434821680954320282657975477573, 10.16710118503500451637999826842, 11.48539989636278238051897861338

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