Properties

Label 2-507-3.2-c0-0-0
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $0.253025$
Root an. cond. $0.503016$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 2·43-s − 48-s − 49-s − 2·61-s + 64-s − 75-s − 2·79-s + 81-s + 100-s − 2·103-s − 108-s + ⋯
L(s)  = 1  − 3-s + 4-s + 9-s − 12-s + 16-s + 25-s − 27-s + 36-s − 2·43-s − 48-s − 49-s − 2·61-s + 64-s − 75-s − 2·79-s + 81-s + 100-s − 2·103-s − 108-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(0.253025\)
Root analytic conductor: \(0.503016\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{507} (170, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8317882095\)
\(L(\frac12)\) \(\approx\) \(0.8317882095\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + T \)
13 \( 1 \)
good2 \( ( 1 - T )( 1 + T ) \)
5 \( ( 1 - T )( 1 + T ) \)
7 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )( 1 + T ) \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( 1 + T^{2} \)
37 \( 1 + T^{2} \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 + T )^{2} \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( ( 1 - T )( 1 + T ) \)
61 \( ( 1 + T )^{2} \)
67 \( 1 + T^{2} \)
71 \( ( 1 - T )( 1 + T ) \)
73 \( 1 + T^{2} \)
79 \( ( 1 + T )^{2} \)
83 \( ( 1 - T )( 1 + T ) \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( 1 + T^{2} \)
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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

−11.13367580834881952500247662593, −10.49163999321511295089647941292, −9.651633738726069267537770192589, −8.282933462069901628483253293923, −7.20996448133866159023524376902, −6.55691982982863666494491758146, −5.67248665125553439266503010963, −4.63712375411729631409827007592, −3.16476648458228068363240929490, −1.59349101491784544382124003930, 1.59349101491784544382124003930, 3.16476648458228068363240929490, 4.63712375411729631409827007592, 5.67248665125553439266503010963, 6.55691982982863666494491758146, 7.20996448133866159023524376902, 8.282933462069901628483253293923, 9.651633738726069267537770192589, 10.49163999321511295089647941292, 11.13367580834881952500247662593

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