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Properties

Label 2-507-507.482-c0-0-0
Degree $2$
Conductor $507$
Sign $0.966 - 0.257i$
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  + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (1.00 − 0.527i)7-s + (−0.354 + 0.935i)9-s + (0.120 − 0.992i)12-s + 13-s + (0.120 + 0.992i)16-s − 0.709·19-s + (1.00 + 0.527i)21-s + (−0.970 − 0.239i)25-s + (−0.970 + 0.239i)27-s + (−1.10 − 0.271i)28-s + (1.45 − 0.358i)31-s + (0.885 − 0.464i)36-s + (−1.71 + 0.423i)37-s + ⋯
L(s)  = 1  + (0.568 + 0.822i)3-s + (−0.748 − 0.663i)4-s + (1.00 − 0.527i)7-s + (−0.354 + 0.935i)9-s + (0.120 − 0.992i)12-s + 13-s + (0.120 + 0.992i)16-s − 0.709·19-s + (1.00 + 0.527i)21-s + (−0.970 − 0.239i)25-s + (−0.970 + 0.239i)27-s + (−1.10 − 0.271i)28-s + (1.45 − 0.358i)31-s + (0.885 − 0.464i)36-s + (−1.71 + 0.423i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.966 - 0.257i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.966 - 0.257i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (482, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.966 - 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9757514977\)
\(L(\frac12)\) \(\approx\) \(0.9757514977\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.568 - 0.822i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.748 + 0.663i)T^{2} \)
5 \( 1 + (0.970 + 0.239i)T^{2} \)
7 \( 1 + (-1.00 + 0.527i)T + (0.568 - 0.822i)T^{2} \)
11 \( 1 + (0.748 - 0.663i)T^{2} \)
17 \( 1 + (-0.568 + 0.822i)T^{2} \)
19 \( 1 + 0.709T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.748 + 0.663i)T^{2} \)
31 \( 1 + (-1.45 + 0.358i)T + (0.885 - 0.464i)T^{2} \)
37 \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \)
41 \( 1 + (0.354 - 0.935i)T^{2} \)
43 \( 1 + (1.71 + 0.423i)T + (0.885 + 0.464i)T^{2} \)
47 \( 1 + (-0.120 + 0.992i)T^{2} \)
53 \( 1 + (-0.568 + 0.822i)T^{2} \)
59 \( 1 + (0.970 + 0.239i)T^{2} \)
61 \( 1 + (1.32 + 0.695i)T + (0.568 + 0.822i)T^{2} \)
67 \( 1 + (-1.45 + 1.28i)T + (0.120 - 0.992i)T^{2} \)
71 \( 1 + (0.354 - 0.935i)T^{2} \)
73 \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \)
79 \( 1 + (0.850 - 0.753i)T + (0.120 - 0.992i)T^{2} \)
83 \( 1 + (0.354 + 0.935i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (-0.241 - 1.98i)T + (-0.970 + 0.239i)T^{2} \)
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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

−10.80438209196802019907763152749, −10.32998071323027854672039554648, −9.432510888571776696279850928136, −8.435573016860440973368035175127, −8.069732727543898072601970698020, −6.42597248050682178716354827897, −5.23168012125239740154015395219, −4.47542255676939500464166491041, −3.62170539929844476000256089837, −1.75282086124197661285480170356, 1.72151642166313286499722722320, 3.13961411403449164315588176809, 4.24525136418783801095203076729, 5.47037790810535737381318739018, 6.65313079928672132616393376007, 7.77700083948033595135860928335, 8.489150418880300317593314317588, 8.807900308384147230436684203622, 10.06246790432902927227444062778, 11.43472419880545319000565834861

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