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Properties

Label 2-507-507.311-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 + (0.764 − 1.45i)7-s + (−0.354 − 0.935i)9-s + (0.120 + 0.992i)12-s − 13-s + (0.120 − 0.992i)16-s + 1.87i·19-s + (0.764 + 1.45i)21-s + (0.970 − 0.239i)25-s + (0.970 + 0.239i)27-s + (−0.393 − 1.59i)28-s + (−0.317 + 1.28i)31-s + (−0.885 − 0.464i)36-s + (−0.222 + 0.902i)37-s + ⋯
L(s)  = 1  + (−0.568 + 0.822i)3-s + (0.748 − 0.663i)4-s + (0.764 − 1.45i)7-s + (−0.354 − 0.935i)9-s + (0.120 + 0.992i)12-s − 13-s + (0.120 − 0.992i)16-s + 1.87i·19-s + (0.764 + 1.45i)21-s + (0.970 − 0.239i)25-s + (0.970 + 0.239i)27-s + (−0.393 − 1.59i)28-s + (−0.317 + 1.28i)31-s + (−0.885 − 0.464i)36-s + (−0.222 + 0.902i)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: $\chi_{507} (311, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.966 + 0.257i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8900384347\)
\(L(\frac12)\) \(\approx\) \(0.8900384347\)
\(L(1)\) 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 \( 1 + (-0.764 + 1.45i)T + (-0.568 - 0.822i)T^{2} \)
11 \( 1 + (-0.748 - 0.663i)T^{2} \)
17 \( 1 + (-0.568 - 0.822i)T^{2} \)
19 \( 1 - 1.87iT - T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.748 - 0.663i)T^{2} \)
31 \( 1 + (0.317 - 1.28i)T + (-0.885 - 0.464i)T^{2} \)
37 \( 1 + (0.222 - 0.902i)T + (-0.885 - 0.464i)T^{2} \)
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 \( 1 + (0.317 - 0.358i)T + (-0.120 - 0.992i)T^{2} \)
71 \( 1 + (-0.354 - 0.935i)T^{2} \)
73 \( 1 + (1.85 + 0.704i)T + (0.748 + 0.663i)T^{2} \)
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 \( 1 + (0.970 + 0.239i)T^{2} \)
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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

−10.90157201716920340408583528677, −10.20545719213500355597488953701, −9.919476358840290152908865377144, −8.362708359199856020850637415101, −7.27249475810723835332002164007, −6.51166405762621893286861316998, −5.30463121743859561569589882609, −4.59661067459612460804891935942, −3.36025413044509201268708109416, −1.43324445884718662899362517414, 2.07047876874325676909310929285, 2.74745224468313176022954662038, 4.80507239839403455328360497193, 5.61886649648684827920630962354, 6.74318774234504262280380690003, 7.41456871370388852675905041171, 8.354967047804767533762226111102, 9.124260848467400453326586375966, 10.66043052167488351786541321215, 11.62601581172869759449349974517

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