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Properties

Label 2-507-507.443-c0-0-0
Degree $2$
Conductor $507$
Sign $0.848 + 0.529i$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.748 + 0.663i)3-s + (−0.970 − 0.239i)4-s + (0.530 − 1.39i)7-s + (0.120 − 0.992i)9-s + (0.885 − 0.464i)12-s + 13-s + (0.885 + 0.464i)16-s + 0.241·19-s + (0.530 + 1.39i)21-s + (0.568 − 0.822i)25-s + (0.568 + 0.822i)27-s + (−0.850 + 1.23i)28-s + (−1.10 − 1.59i)31-s + (−0.354 + 0.935i)36-s + (−0.402 − 0.583i)37-s + ⋯
L(s)  = 1  + (−0.748 + 0.663i)3-s + (−0.970 − 0.239i)4-s + (0.530 − 1.39i)7-s + (0.120 − 0.992i)9-s + (0.885 − 0.464i)12-s + 13-s + (0.885 + 0.464i)16-s + 0.241·19-s + (0.530 + 1.39i)21-s + (0.568 − 0.822i)25-s + (0.568 + 0.822i)27-s + (−0.850 + 1.23i)28-s + (−1.10 − 1.59i)31-s + (−0.354 + 0.935i)36-s + (−0.402 − 0.583i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.848 + 0.529i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.848 + 0.529i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.848 + 0.529i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.6032935877\)
\(L(\frac12)\) \(\approx\) \(0.6032935877\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.748 - 0.663i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.970 + 0.239i)T^{2} \)
5 \( 1 + (-0.568 + 0.822i)T^{2} \)
7 \( 1 + (-0.530 + 1.39i)T + (-0.748 - 0.663i)T^{2} \)
11 \( 1 + (0.970 - 0.239i)T^{2} \)
17 \( 1 + (0.748 + 0.663i)T^{2} \)
19 \( 1 - 0.241T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.970 + 0.239i)T^{2} \)
31 \( 1 + (1.10 + 1.59i)T + (-0.354 + 0.935i)T^{2} \)
37 \( 1 + (0.402 + 0.583i)T + (-0.354 + 0.935i)T^{2} \)
41 \( 1 + (-0.120 + 0.992i)T^{2} \)
43 \( 1 + (0.402 - 0.583i)T + (-0.354 - 0.935i)T^{2} \)
47 \( 1 + (-0.885 + 0.464i)T^{2} \)
53 \( 1 + (0.748 + 0.663i)T^{2} \)
59 \( 1 + (-0.568 + 0.822i)T^{2} \)
61 \( 1 + (-0.688 - 1.81i)T + (-0.748 + 0.663i)T^{2} \)
67 \( 1 + (1.10 - 0.271i)T + (0.885 - 0.464i)T^{2} \)
71 \( 1 + (-0.120 + 0.992i)T^{2} \)
73 \( 1 + (-0.213 - 1.75i)T + (-0.970 + 0.239i)T^{2} \)
79 \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \)
83 \( 1 + (-0.120 - 0.992i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-1.77 - 0.929i)T + (0.568 + 0.822i)T^{2} \)
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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

−10.82277580867878914423584756260, −10.32199008831496025704577569005, −9.436123968673476351774777284125, −8.519356570194236414654998474335, −7.43973093536313168354502539165, −6.23799173345218703880777741417, −5.26817071256981804897065643003, −4.27504845725265304023132599440, −3.75342375159133429946796502683, −0.989238509802724029433676455237, 1.64224947615562114443668028908, 3.34601220617821666548728804122, 4.96572210899860641809723455260, 5.42213649073172144712534604425, 6.49208135256462039196210832684, 7.72294450777560374979528281628, 8.601997812539652590675261906241, 9.135051002039058478767870974064, 10.48842959804684220804525554685, 11.35410402160879281207822208702

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