Properties

Label 2-507-1.1-c1-0-5
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.04·2-s − 3-s + 2.19·4-s + 3.35·5-s + 2.04·6-s + 2.24·7-s − 0.405·8-s + 9-s − 6.87·10-s + 4.93·11-s − 2.19·12-s − 4.60·14-s − 3.35·15-s − 3.56·16-s + 0.911·17-s − 2.04·18-s − 3.80·19-s + 7.37·20-s − 2.24·21-s − 10.1·22-s + 2.02·23-s + 0.405·24-s + 6.26·25-s − 27-s + 4.93·28-s − 3.93·29-s + 6.87·30-s + ⋯
L(s)  = 1  − 1.44·2-s − 0.577·3-s + 1.09·4-s + 1.50·5-s + 0.836·6-s + 0.849·7-s − 0.143·8-s + 0.333·9-s − 2.17·10-s + 1.48·11-s − 0.634·12-s − 1.23·14-s − 0.866·15-s − 0.891·16-s + 0.221·17-s − 0.482·18-s − 0.872·19-s + 1.64·20-s − 0.490·21-s − 2.15·22-s + 0.422·23-s + 0.0828·24-s + 1.25·25-s − 0.192·27-s + 0.933·28-s − 0.731·29-s + 1.25·30-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: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8884190576\)
\(L(\frac12)\) \(\approx\) \(0.8884190576\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.04T + 2T^{2} \)
5 \( 1 - 3.35T + 5T^{2} \)
7 \( 1 - 2.24T + 7T^{2} \)
11 \( 1 - 4.93T + 11T^{2} \)
17 \( 1 - 0.911T + 17T^{2} \)
19 \( 1 + 3.80T + 19T^{2} \)
23 \( 1 - 2.02T + 23T^{2} \)
29 \( 1 + 3.93T + 29T^{2} \)
31 \( 1 + 8.82T + 31T^{2} \)
37 \( 1 - 8.80T + 37T^{2} \)
41 \( 1 - 6.93T + 41T^{2} \)
43 \( 1 + 2.28T + 43T^{2} \)
47 \( 1 - 3.80T + 47T^{2} \)
53 \( 1 - 0.542T + 53T^{2} \)
59 \( 1 + 4.71T + 59T^{2} \)
61 \( 1 - 3.67T + 61T^{2} \)
67 \( 1 + 1.52T + 67T^{2} \)
71 \( 1 - 2.37T + 71T^{2} \)
73 \( 1 - 7.41T + 73T^{2} \)
79 \( 1 + 3.74T + 79T^{2} \)
83 \( 1 - 2.30T + 83T^{2} \)
89 \( 1 + 10.0T + 89T^{2} \)
97 \( 1 + 16.1T + 97T^{2} \)
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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

−10.94693369393660235677203284829, −9.718590993472648888920182265333, −9.365669293112238223844295430788, −8.506509273980416654680785872432, −7.34056233531221372370716228875, −6.45385532909737390433362921761, −5.58778236919420814783250669715, −4.31631134428115154031449675131, −2.05843232776131367445206750053, −1.25195005858148234988264507219, 1.25195005858148234988264507219, 2.05843232776131367445206750053, 4.31631134428115154031449675131, 5.58778236919420814783250669715, 6.45385532909737390433362921761, 7.34056233531221372370716228875, 8.506509273980416654680785872432, 9.365669293112238223844295430788, 9.718590993472648888920182265333, 10.94693369393660235677203284829

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