Properties

Label 2-507-1.1-c1-0-7
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  + 0.445·2-s + 3-s − 1.80·4-s − 0.246·5-s + 0.445·6-s + 1.75·7-s − 1.69·8-s + 9-s − 0.109·10-s + 5.65·11-s − 1.80·12-s + 0.780·14-s − 0.246·15-s + 2.85·16-s − 3.80·17-s + 0.445·18-s + 5.58·19-s + 0.445·20-s + 1.75·21-s + 2.51·22-s + 8.34·23-s − 1.69·24-s − 4.93·25-s + 27-s − 3.15·28-s − 5.93·29-s − 0.109·30-s + ⋯
L(s)  = 1  + 0.314·2-s + 0.577·3-s − 0.900·4-s − 0.110·5-s + 0.181·6-s + 0.662·7-s − 0.598·8-s + 0.333·9-s − 0.0347·10-s + 1.70·11-s − 0.520·12-s + 0.208·14-s − 0.0637·15-s + 0.712·16-s − 0.922·17-s + 0.104·18-s + 1.28·19-s + 0.0995·20-s + 0.382·21-s + 0.536·22-s + 1.74·23-s − 0.345·24-s − 0.987·25-s + 0.192·27-s − 0.596·28-s − 1.10·29-s − 0.0200·30-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

\(L(1)\) \(\approx\) \(1.851015433\)
\(L(\frac12)\) \(\approx\) \(1.851015433\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.445T + 2T^{2} \)
5 \( 1 + 0.246T + 5T^{2} \)
7 \( 1 - 1.75T + 7T^{2} \)
11 \( 1 - 5.65T + 11T^{2} \)
17 \( 1 + 3.80T + 17T^{2} \)
19 \( 1 - 5.58T + 19T^{2} \)
23 \( 1 - 8.34T + 23T^{2} \)
29 \( 1 + 5.93T + 29T^{2} \)
31 \( 1 - 5.26T + 31T^{2} \)
37 \( 1 - 3.19T + 37T^{2} \)
41 \( 1 - 0.445T + 41T^{2} \)
43 \( 1 - 1.71T + 43T^{2} \)
47 \( 1 + 6.73T + 47T^{2} \)
53 \( 1 + 1.06T + 53T^{2} \)
59 \( 1 + 13.7T + 59T^{2} \)
61 \( 1 + 8.51T + 61T^{2} \)
67 \( 1 - 5.96T + 67T^{2} \)
71 \( 1 + 5.71T + 71T^{2} \)
73 \( 1 - 7.35T + 73T^{2} \)
79 \( 1 - 4.45T + 79T^{2} \)
83 \( 1 + 10.1T + 83T^{2} \)
89 \( 1 - 0.137T + 89T^{2} \)
97 \( 1 + 13.6T + 97T^{2} \)
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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

−11.12002966754360133829965115475, −9.561021900356378255153181199683, −9.262435606503044850961956068134, −8.366793768393057078528770929787, −7.39260193239174232205051567090, −6.25426652103977034272802957201, −4.97181311706309888084725703013, −4.17826828325452043239009646316, −3.20298903581918346857264570917, −1.36618448393371556817901132706, 1.36618448393371556817901132706, 3.20298903581918346857264570917, 4.17826828325452043239009646316, 4.97181311706309888084725703013, 6.25426652103977034272802957201, 7.39260193239174232205051567090, 8.366793768393057078528770929787, 9.262435606503044850961956068134, 9.561021900356378255153181199683, 11.12002966754360133829965115475

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