Properties

Label 2-507-1.1-c3-0-55
Degree $2$
Conductor $507$
Sign $-1$
Analytic cond. $29.9139$
Root an. cond. $5.46936$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s + 3·3-s + 4-s + 9·5-s − 9·6-s − 2·7-s + 21·8-s + 9·9-s − 27·10-s − 30·11-s + 3·12-s + 6·14-s + 27·15-s − 71·16-s − 111·17-s − 27·18-s + 46·19-s + 9·20-s − 6·21-s + 90·22-s − 6·23-s + 63·24-s − 44·25-s + 27·27-s − 2·28-s − 105·29-s − 81·30-s + ⋯
L(s)  = 1  − 1.06·2-s + 0.577·3-s + 1/8·4-s + 0.804·5-s − 0.612·6-s − 0.107·7-s + 0.928·8-s + 1/3·9-s − 0.853·10-s − 0.822·11-s + 0.0721·12-s + 0.114·14-s + 0.464·15-s − 1.10·16-s − 1.58·17-s − 0.353·18-s + 0.555·19-s + 0.100·20-s − 0.0623·21-s + 0.872·22-s − 0.0543·23-s + 0.535·24-s − 0.351·25-s + 0.192·27-s − 0.0134·28-s − 0.672·29-s − 0.492·30-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: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 507,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - p T \)
13 \( 1 \)
good2 \( 1 + 3 T + p^{3} T^{2} \)
5 \( 1 - 9 T + p^{3} T^{2} \)
7 \( 1 + 2 T + p^{3} T^{2} \)
11 \( 1 + 30 T + p^{3} T^{2} \)
17 \( 1 + 111 T + p^{3} T^{2} \)
19 \( 1 - 46 T + p^{3} T^{2} \)
23 \( 1 + 6 T + p^{3} T^{2} \)
29 \( 1 + 105 T + p^{3} T^{2} \)
31 \( 1 - 100 T + p^{3} T^{2} \)
37 \( 1 + 17 T + p^{3} T^{2} \)
41 \( 1 - 231 T + p^{3} T^{2} \)
43 \( 1 + 514 T + p^{3} T^{2} \)
47 \( 1 - 162 T + p^{3} T^{2} \)
53 \( 1 - 639 T + p^{3} T^{2} \)
59 \( 1 + 600 T + p^{3} T^{2} \)
61 \( 1 - 233 T + p^{3} T^{2} \)
67 \( 1 + 926 T + p^{3} T^{2} \)
71 \( 1 - 930 T + p^{3} T^{2} \)
73 \( 1 - 253 T + p^{3} T^{2} \)
79 \( 1 + 1324 T + p^{3} T^{2} \)
83 \( 1 + 810 T + p^{3} T^{2} \)
89 \( 1 + 498 T + p^{3} T^{2} \)
97 \( 1 + 14 p T + p^{3} T^{2} \)
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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

−9.866217548633004449106839787889, −9.194830966383675184165454641570, −8.451763235758352796712708115192, −7.60890377638021368282601318689, −6.65058606664874317616957796889, −5.35056723379727738465505340985, −4.23016663879819715368972859203, −2.62537180424150233080846076205, −1.61839528529427016298760100429, 0, 1.61839528529427016298760100429, 2.62537180424150233080846076205, 4.23016663879819715368972859203, 5.35056723379727738465505340985, 6.65058606664874317616957796889, 7.60890377638021368282601318689, 8.451763235758352796712708115192, 9.194830966383675184165454641570, 9.866217548633004449106839787889

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