Properties

Label 2-10580-1.1-c1-0-2
Degree $2$
Conductor $10580$
Sign $1$
Analytic cond. $84.4817$
Root an. cond. $9.19139$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·3-s + 5-s − 2·7-s + 9-s + 2·13-s − 2·15-s + 6·17-s + 4·19-s + 4·21-s + 25-s + 4·27-s + 6·29-s − 4·31-s − 2·35-s − 2·37-s − 4·39-s + 6·41-s + 10·43-s + 45-s − 6·47-s − 3·49-s − 12·51-s + 6·53-s − 8·57-s + 12·59-s − 2·61-s − 2·63-s + ⋯
L(s)  = 1  − 1.15·3-s + 0.447·5-s − 0.755·7-s + 1/3·9-s + 0.554·13-s − 0.516·15-s + 1.45·17-s + 0.917·19-s + 0.872·21-s + 1/5·25-s + 0.769·27-s + 1.11·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.640·39-s + 0.937·41-s + 1.52·43-s + 0.149·45-s − 0.875·47-s − 3/7·49-s − 1.68·51-s + 0.824·53-s − 1.05·57-s + 1.56·59-s − 0.256·61-s − 0.251·63-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 10580 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10580 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(10580\)    =    \(2^{2} \cdot 5 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(84.4817\)
Root analytic conductor: \(9.19139\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 10580,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.422588671\)
\(L(\frac12)\) \(\approx\) \(1.422588671\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 - T \)
23 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 2 T + p T^{2} \)
11 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 10 T + p T^{2} \)
47 \( 1 + 6 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−16.40093088660464, −16.19793502030992, −15.78432560702860, −14.72545500579591, −14.26012553415601, −13.70841640058731, −12.86924341476825, −12.59052768606086, −11.79986299379600, −11.51271413248493, −10.61237205368062, −10.26878264807607, −9.634470388800409, −9.026864902926892, −8.226596894071287, −7.415847606801096, −6.796228802319020, −6.074436654034723, −5.673640184666111, −5.177585572393149, −4.229812344220990, −3.333102984084275, −2.713258903586918, −1.366468923151831, −0.6565854589594138, 0.6565854589594138, 1.366468923151831, 2.713258903586918, 3.333102984084275, 4.229812344220990, 5.177585572393149, 5.673640184666111, 6.074436654034723, 6.796228802319020, 7.415847606801096, 8.226596894071287, 9.026864902926892, 9.634470388800409, 10.26878264807607, 10.61237205368062, 11.51271413248493, 11.79986299379600, 12.59052768606086, 12.86924341476825, 13.70841640058731, 14.26012553415601, 14.72545500579591, 15.78432560702860, 16.19793502030992, 16.40093088660464

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