Properties

Label 2-507-1.1-c1-0-18
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  + 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: \(0\)
Selberg data: \((2,\ 507,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.427037441\)
\(L(\frac12)\) \(\approx\) \(3.427037441\)
\(L(\frac{3}{2})\) 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

−11.15328047701435280075041460083, −10.06293087689440780133977725130, −9.155827761908500415080014517113, −8.025260521216847421627443907545, −7.35323204584666956427716043452, −5.71329151041676788344608061322, −5.29296223508686036055147160322, −4.28595217044612382967827364550, −3.01693015539831466587929705758, −1.98368280574573453517805604990, 1.98368280574573453517805604990, 3.01693015539831466587929705758, 4.28595217044612382967827364550, 5.29296223508686036055147160322, 5.71329151041676788344608061322, 7.35323204584666956427716043452, 8.025260521216847421627443907545, 9.155827761908500415080014517113, 10.06293087689440780133977725130, 11.15328047701435280075041460083

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