Properties

Label 2-507-1.1-c3-0-25
Degree $2$
Conductor $507$
Sign $1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82·2-s + 3·3-s − 0.0423·4-s − 3.41·5-s − 8.46·6-s + 13.3·7-s + 22.6·8-s + 9·9-s + 9.62·10-s + 35.4·11-s − 0.127·12-s − 37.6·14-s − 10.2·15-s − 63.6·16-s + 69.6·17-s − 25.3·18-s + 12.4·19-s + 0.144·20-s + 40.0·21-s − 100.·22-s − 126.·23-s + 68.0·24-s − 113.·25-s + 27·27-s − 0.565·28-s − 179.·29-s + 28.8·30-s + ⋯
L(s)  = 1  − 0.997·2-s + 0.577·3-s − 0.00529·4-s − 0.305·5-s − 0.575·6-s + 0.720·7-s + 1.00·8-s + 0.333·9-s + 0.304·10-s + 0.971·11-s − 0.00305·12-s − 0.718·14-s − 0.176·15-s − 0.994·16-s + 0.993·17-s − 0.332·18-s + 0.149·19-s + 0.00161·20-s + 0.415·21-s − 0.969·22-s − 1.14·23-s + 0.578·24-s − 0.906·25-s + 0.192·27-s − 0.00381·28-s − 1.14·29-s + 0.175·30-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(507\)    =    \(3 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(29.9139\)
Root analytic conductor: \(5.46936\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(1.433834885\)
\(L(\frac12)\) \(\approx\) \(1.433834885\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 3T \)
13 \( 1 \)
good2 \( 1 + 2.82T + 8T^{2} \)
5 \( 1 + 3.41T + 125T^{2} \)
7 \( 1 - 13.3T + 343T^{2} \)
11 \( 1 - 35.4T + 1.33e3T^{2} \)
17 \( 1 - 69.6T + 4.91e3T^{2} \)
19 \( 1 - 12.4T + 6.85e3T^{2} \)
23 \( 1 + 126.T + 1.21e4T^{2} \)
29 \( 1 + 179.T + 2.43e4T^{2} \)
31 \( 1 - 255.T + 2.97e4T^{2} \)
37 \( 1 - 207.T + 5.06e4T^{2} \)
41 \( 1 + 117.T + 6.89e4T^{2} \)
43 \( 1 - 553.T + 7.95e4T^{2} \)
47 \( 1 - 62.9T + 1.03e5T^{2} \)
53 \( 1 + 147.T + 1.48e5T^{2} \)
59 \( 1 - 274.T + 2.05e5T^{2} \)
61 \( 1 - 603.T + 2.26e5T^{2} \)
67 \( 1 + 741.T + 3.00e5T^{2} \)
71 \( 1 - 572.T + 3.57e5T^{2} \)
73 \( 1 - 26.7T + 3.89e5T^{2} \)
79 \( 1 + 207.T + 4.93e5T^{2} \)
83 \( 1 + 1.03e3T + 5.71e5T^{2} \)
89 \( 1 - 1.22e3T + 7.04e5T^{2} \)
97 \( 1 - 1.79e3T + 9.12e5T^{2} \)
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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

−10.12914367568423894543786746331, −9.574353446708088195113328959172, −8.682633798270794769287075387494, −7.915813616726552132590989900321, −7.42401615283105322594481860504, −5.97975731705937039381157170005, −4.55437658177734135468961668472, −3.72720435663912562660599416850, −1.99532150518274444738183658580, −0.890485594625183678833186140230, 0.890485594625183678833186140230, 1.99532150518274444738183658580, 3.72720435663912562660599416850, 4.55437658177734135468961668472, 5.97975731705937039381157170005, 7.42401615283105322594481860504, 7.915813616726552132590989900321, 8.682633798270794769287075387494, 9.574353446708088195113328959172, 10.12914367568423894543786746331

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