Properties

Label 2-507-1.1-c1-0-14
Degree $2$
Conductor $507$
Sign $-1$
Analytic cond. $4.04841$
Root an. cond. $2.01206$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 1.80·2-s + 3-s + 1.24·4-s − 1.44·5-s − 1.80·6-s − 3.44·7-s + 1.35·8-s + 9-s + 2.60·10-s + 5.18·11-s + 1.24·12-s + 6.20·14-s − 1.44·15-s − 4.93·16-s − 0.753·17-s − 1.80·18-s − 7.96·19-s − 1.80·20-s − 3.44·21-s − 9.34·22-s − 2.82·23-s + 1.35·24-s − 2.91·25-s + 27-s − 4.29·28-s − 3.91·29-s + 2.60·30-s + ⋯
L(s)  = 1  − 1.27·2-s + 0.577·3-s + 0.623·4-s − 0.646·5-s − 0.735·6-s − 1.30·7-s + 0.479·8-s + 0.333·9-s + 0.823·10-s + 1.56·11-s + 0.359·12-s + 1.65·14-s − 0.373·15-s − 1.23·16-s − 0.182·17-s − 0.424·18-s − 1.82·19-s − 0.402·20-s − 0.751·21-s − 1.99·22-s − 0.589·23-s + 0.276·24-s − 0.582·25-s + 0.192·27-s − 0.811·28-s − 0.726·29-s + 0.475·30-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(4.04841\)
Root analytic conductor: \(2.01206\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{507} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.80T + 2T^{2} \)
5 \( 1 + 1.44T + 5T^{2} \)
7 \( 1 + 3.44T + 7T^{2} \)
11 \( 1 - 5.18T + 11T^{2} \)
17 \( 1 + 0.753T + 17T^{2} \)
19 \( 1 + 7.96T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 + 3.91T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 6.24T + 37T^{2} \)
41 \( 1 + 1.80T + 41T^{2} \)
43 \( 1 + 7.09T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 + 3.08T + 53T^{2} \)
59 \( 1 + 1.87T + 59T^{2} \)
61 \( 1 - 3.34T + 61T^{2} \)
67 \( 1 - 4.54T + 67T^{2} \)
71 \( 1 - 9.11T + 71T^{2} \)
73 \( 1 + 2.95T + 73T^{2} \)
79 \( 1 + 9.43T + 79T^{2} \)
83 \( 1 - 6.46T + 83T^{2} \)
89 \( 1 + 1.15T + 89T^{2} \)
97 \( 1 + 8.65T + 97T^{2} \)
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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

−10.04622159425429106782801160423, −9.545318827431046671283032706021, −8.688382530171444118859210317137, −8.134379072855960489066306656933, −6.89865178120035937609479784441, −6.42142572689273810047465156245, −4.32191340047312199842370424717, −3.52848953311962123409972105103, −1.85376290648341535326764824068, 0, 1.85376290648341535326764824068, 3.52848953311962123409972105103, 4.32191340047312199842370424717, 6.42142572689273810047465156245, 6.89865178120035937609479784441, 8.134379072855960489066306656933, 8.688382530171444118859210317137, 9.545318827431046671283032706021, 10.04622159425429106782801160423

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