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Properties

Label 2-507-507.326-c0-0-0
Degree $2$
Conductor $507$
Sign $0.230 - 0.973i$
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  + (0.885 + 0.464i)3-s + (−0.354 + 0.935i)4-s + (−1.71 + 0.423i)7-s + (0.568 + 0.822i)9-s + (−0.748 + 0.663i)12-s + 13-s + (−0.748 − 0.663i)16-s + 1.13·19-s + (−1.71 − 0.423i)21-s + (0.120 − 0.992i)25-s + (0.120 + 0.992i)27-s + (0.213 − 1.75i)28-s + (−0.0854 − 0.704i)31-s + (−0.970 + 0.239i)36-s + (−0.234 − 1.92i)37-s + ⋯
L(s)  = 1  + (0.885 + 0.464i)3-s + (−0.354 + 0.935i)4-s + (−1.71 + 0.423i)7-s + (0.568 + 0.822i)9-s + (−0.748 + 0.663i)12-s + 13-s + (−0.748 − 0.663i)16-s + 1.13·19-s + (−1.71 − 0.423i)21-s + (0.120 − 0.992i)25-s + (0.120 + 0.992i)27-s + (0.213 − 1.75i)28-s + (−0.0854 − 0.704i)31-s + (−0.970 + 0.239i)36-s + (−0.234 − 1.92i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $0.230 - 0.973i$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (326, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 0.230 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9537894758\)
\(L(\frac12)\) \(\approx\) \(0.9537894758\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.885 - 0.464i)T \)
13 \( 1 - T \)
good2 \( 1 + (0.354 - 0.935i)T^{2} \)
5 \( 1 + (-0.120 + 0.992i)T^{2} \)
7 \( 1 + (1.71 - 0.423i)T + (0.885 - 0.464i)T^{2} \)
11 \( 1 + (0.354 + 0.935i)T^{2} \)
17 \( 1 + (-0.885 + 0.464i)T^{2} \)
19 \( 1 - 1.13T + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 + (0.354 - 0.935i)T^{2} \)
31 \( 1 + (0.0854 + 0.704i)T + (-0.970 + 0.239i)T^{2} \)
37 \( 1 + (0.234 + 1.92i)T + (-0.970 + 0.239i)T^{2} \)
41 \( 1 + (-0.568 - 0.822i)T^{2} \)
43 \( 1 + (0.234 - 1.92i)T + (-0.970 - 0.239i)T^{2} \)
47 \( 1 + (0.748 - 0.663i)T^{2} \)
53 \( 1 + (-0.885 + 0.464i)T^{2} \)
59 \( 1 + (-0.120 + 0.992i)T^{2} \)
61 \( 1 + (-0.688 - 0.169i)T + (0.885 + 0.464i)T^{2} \)
67 \( 1 + (0.0854 + 0.225i)T + (-0.748 + 0.663i)T^{2} \)
71 \( 1 + (-0.568 - 0.822i)T^{2} \)
73 \( 1 + (0.850 - 1.23i)T + (-0.354 - 0.935i)T^{2} \)
79 \( 1 + (0.627 + 1.65i)T + (-0.748 + 0.663i)T^{2} \)
83 \( 1 + (-0.568 + 0.822i)T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + (1.49 + 1.32i)T + (0.120 + 0.992i)T^{2} \)
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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

−11.32901045413546249672000156857, −10.08771210882208208304438159586, −9.392350098900823583268868996831, −8.810005404269206204484537768537, −7.890171873869982775151186014960, −6.92329916408975075306318686841, −5.76446659657228779238834700844, −4.23415641760377206151041882263, −3.41294836794802807035578756306, −2.68365552622320407843051042059, 1.27214755978884197280193647374, 3.06878854122491148788811491712, 3.86532289827420228117707333591, 5.46735966545717803437908886026, 6.53020789530577721280104107433, 7.07751691670734498967648711835, 8.463867004560601106905136506987, 9.286647831073475646917694499093, 9.851631448355926480206869026054, 10.63186967588245656129677034683

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