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Properties

Label 2-507-507.428-c0-0-0
Degree $2$
Conductor $507$
Sign $0.956 + 0.293i$
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  + (0.970 + 0.239i)3-s + (−0.568 − 0.822i)4-s + (0.475 − 0.0576i)7-s + (0.885 + 0.464i)9-s + (−0.354 − 0.935i)12-s − 13-s + (−0.354 + 0.935i)16-s − 0.929i·19-s + (0.475 + 0.0576i)21-s + (0.748 − 0.663i)25-s + (0.748 + 0.663i)27-s + (−0.317 − 0.358i)28-s + (−1.09 + 1.23i)31-s + (−0.120 − 0.992i)36-s + (−1.31 + 1.48i)37-s + ⋯
L(s)  = 1  + (0.970 + 0.239i)3-s + (−0.568 − 0.822i)4-s + (0.475 − 0.0576i)7-s + (0.885 + 0.464i)9-s + (−0.354 − 0.935i)12-s − 13-s + (−0.354 + 0.935i)16-s − 0.929i·19-s + (0.475 + 0.0576i)21-s + (0.748 − 0.663i)25-s + (0.748 + 0.663i)27-s + (−0.317 − 0.358i)28-s + (−1.09 + 1.23i)31-s + (−0.120 − 0.992i)36-s + (−1.31 + 1.48i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.956 + 0.293i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.956 + 0.293i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (428, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.956 + 0.293i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.074890974\)
\(L(\frac12)\) \(\approx\) \(1.074890974\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.970 - 0.239i)T \)
13 \( 1 + T \)
good2 \( 1 + (0.568 + 0.822i)T^{2} \)
5 \( 1 + (-0.748 + 0.663i)T^{2} \)
7 \( 1 + (-0.475 + 0.0576i)T + (0.970 - 0.239i)T^{2} \)
11 \( 1 + (0.568 - 0.822i)T^{2} \)
17 \( 1 + (0.970 - 0.239i)T^{2} \)
19 \( 1 + 0.929iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (-0.568 - 0.822i)T^{2} \)
31 \( 1 + (1.09 - 1.23i)T + (-0.120 - 0.992i)T^{2} \)
37 \( 1 + (1.31 - 1.48i)T + (-0.120 - 0.992i)T^{2} \)
41 \( 1 + (0.885 + 0.464i)T^{2} \)
43 \( 1 + (0.180 - 0.159i)T + (0.120 - 0.992i)T^{2} \)
47 \( 1 + (-0.354 - 0.935i)T^{2} \)
53 \( 1 + (0.970 - 0.239i)T^{2} \)
59 \( 1 + (-0.748 + 0.663i)T^{2} \)
61 \( 1 + (0.136 - 1.12i)T + (-0.970 - 0.239i)T^{2} \)
67 \( 1 + (1.09 + 0.753i)T + (0.354 + 0.935i)T^{2} \)
71 \( 1 + (0.885 + 0.464i)T^{2} \)
73 \( 1 + (0.869 + 1.65i)T + (-0.568 + 0.822i)T^{2} \)
79 \( 1 + (-1.10 + 1.59i)T + (-0.354 - 0.935i)T^{2} \)
83 \( 1 + (0.885 - 0.464i)T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 + (0.748 + 0.663i)T^{2} \)
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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

−10.67960394919012456628485995686, −10.20618959659429841479846584623, −9.173525563264286385522555740889, −8.692968033847583927887424195656, −7.58600267724131361688311127422, −6.61167787103983025269967285210, −5.03242294261218210610885966816, −4.62311813243305384003239217229, −3.13211183498494831522777939118, −1.73338169664672305746202702984, 2.06549651819913184896167051421, 3.32320028562272911836455812172, 4.23648660512712352343410152084, 5.36939482481818130939916473335, 7.06831158286090431882202437868, 7.65095841720238501088461383925, 8.465670048825532618801684272972, 9.228347463675684275905561484575, 9.991965186764809529485865252128, 11.26638704171439116653868379017

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