Properties

Label 2-507-1.1-c1-0-23
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  + 1.24·2-s + 3-s − 0.445·4-s − 2.80·5-s + 1.24·6-s − 4.80·7-s − 3.04·8-s + 9-s − 3.49·10-s + 1.46·11-s − 0.445·12-s − 5.98·14-s − 2.80·15-s − 2.91·16-s − 2.44·17-s + 1.24·18-s + 2.54·19-s + 1.24·20-s − 4.80·21-s + 1.82·22-s − 3.51·23-s − 3.04·24-s + 2.85·25-s + 27-s + 2.13·28-s + 1.85·29-s − 3.49·30-s + ⋯
L(s)  = 1  + 0.881·2-s + 0.577·3-s − 0.222·4-s − 1.25·5-s + 0.509·6-s − 1.81·7-s − 1.07·8-s + 0.333·9-s − 1.10·10-s + 0.442·11-s − 0.128·12-s − 1.60·14-s − 0.723·15-s − 0.727·16-s − 0.593·17-s + 0.293·18-s + 0.583·19-s + 0.278·20-s − 1.04·21-s + 0.389·22-s − 0.733·23-s − 0.622·24-s + 0.570·25-s + 0.192·27-s + 0.403·28-s + 0.343·29-s − 0.637·30-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 \( 1 - 1.24T + 2T^{2} \)
5 \( 1 + 2.80T + 5T^{2} \)
7 \( 1 + 4.80T + 7T^{2} \)
11 \( 1 - 1.46T + 11T^{2} \)
17 \( 1 + 2.44T + 17T^{2} \)
19 \( 1 - 2.54T + 19T^{2} \)
23 \( 1 + 3.51T + 23T^{2} \)
29 \( 1 - 1.85T + 29T^{2} \)
31 \( 1 + 7.63T + 31T^{2} \)
37 \( 1 + 4.55T + 37T^{2} \)
41 \( 1 - 1.24T + 41T^{2} \)
43 \( 1 - 2.38T + 43T^{2} \)
47 \( 1 - 12.8T + 47T^{2} \)
53 \( 1 + 8.85T + 53T^{2} \)
59 \( 1 - 2.17T + 59T^{2} \)
61 \( 1 + 7.82T + 61T^{2} \)
67 \( 1 + 3.58T + 67T^{2} \)
71 \( 1 + 8.83T + 71T^{2} \)
73 \( 1 + 7.69T + 73T^{2} \)
79 \( 1 + 4.02T + 79T^{2} \)
83 \( 1 + 0.652T + 83T^{2} \)
89 \( 1 - 6.29T + 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
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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

−10.44099667716575540984541779662, −9.328664358982393923814901433342, −8.901916956417484952978620653598, −7.62837194704805766727118585463, −6.73304836595896926602913362769, −5.75526842125667722901185388159, −4.26503047454121388669926092409, −3.69110187342332593466895818269, −2.91060722505024791174012062239, 0, 2.91060722505024791174012062239, 3.69110187342332593466895818269, 4.26503047454121388669926092409, 5.75526842125667722901185388159, 6.73304836595896926602913362769, 7.62837194704805766727118585463, 8.901916956417484952978620653598, 9.328664358982393923814901433342, 10.44099667716575540984541779662

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