Properties

Label 2-507-1.1-c1-0-19
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  + 2.41·2-s + 3-s + 3.82·4-s − 2.82·5-s + 2.41·6-s + 2.82·7-s + 4.41·8-s + 9-s − 6.82·10-s + 2·11-s + 3.82·12-s + 6.82·14-s − 2.82·15-s + 2.99·16-s − 3.65·17-s + 2.41·18-s − 2.82·19-s − 10.8·20-s + 2.82·21-s + 4.82·22-s − 4·23-s + 4.41·24-s + 3.00·25-s + 27-s + 10.8·28-s + 2·29-s − 6.82·30-s + ⋯
L(s)  = 1  + 1.70·2-s + 0.577·3-s + 1.91·4-s − 1.26·5-s + 0.985·6-s + 1.06·7-s + 1.56·8-s + 0.333·9-s − 2.15·10-s + 0.603·11-s + 1.10·12-s + 1.82·14-s − 0.730·15-s + 0.749·16-s − 0.886·17-s + 0.569·18-s − 0.648·19-s − 2.42·20-s + 0.617·21-s + 1.02·22-s − 0.834·23-s + 0.901·24-s + 0.600·25-s + 0.192·27-s + 2.04·28-s + 0.371·29-s − 1.24·30-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

\(L(1)\) \(\approx\) \(3.900004773\)
\(L(\frac12)\) \(\approx\) \(3.900004773\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.41T + 2T^{2} \)
5 \( 1 + 2.82T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
17 \( 1 + 3.65T + 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2T + 29T^{2} \)
31 \( 1 - 6.82T + 31T^{2} \)
37 \( 1 + 3.65T + 37T^{2} \)
41 \( 1 + 10.8T + 41T^{2} \)
43 \( 1 - 9.65T + 43T^{2} \)
47 \( 1 - 0.343T + 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 - 3.65T + 59T^{2} \)
61 \( 1 + 9.31T + 61T^{2} \)
67 \( 1 + 1.17T + 67T^{2} \)
71 \( 1 + 2T + 71T^{2} \)
73 \( 1 + 11.6T + 73T^{2} \)
79 \( 1 - 11.3T + 79T^{2} \)
83 \( 1 - 7.65T + 83T^{2} \)
89 \( 1 + 9.17T + 89T^{2} \)
97 \( 1 - 7.65T + 97T^{2} \)
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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

−11.40636187453181820218716400890, −10.46353179355012938863685529446, −8.804453106396835433251662182094, −8.043318461451174999840626860467, −7.13234674191477123771960679152, −6.17980760568330452251597112081, −4.70363284754302942564370984901, −4.30286503434185049344072381883, −3.36008278962898975441928512649, −2.02541190877468342186149530379, 2.02541190877468342186149530379, 3.36008278962898975441928512649, 4.30286503434185049344072381883, 4.70363284754302942564370984901, 6.17980760568330452251597112081, 7.13234674191477123771960679152, 8.043318461451174999840626860467, 8.804453106396835433251662182094, 10.46353179355012938863685529446, 11.40636187453181820218716400890

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