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Properties

Label 2-59840-1.1-c1-0-9
Degree $2$
Conductor $59840$
Sign $1$
Analytic cond. $477.824$
Root an. cond. $21.8592$
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  + 3-s + 5-s + 4·7-s − 2·9-s − 11-s − 5·13-s + 15-s + 17-s − 19-s + 4·21-s + 25-s − 5·27-s − 9·29-s + 7·31-s − 33-s + 4·35-s − 8·37-s − 5·39-s − 10·43-s − 2·45-s − 3·47-s + 9·49-s + 51-s + 3·53-s − 55-s − 57-s + 9·59-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.447·5-s + 1.51·7-s − 2/3·9-s − 0.301·11-s − 1.38·13-s + 0.258·15-s + 0.242·17-s − 0.229·19-s + 0.872·21-s + 1/5·25-s − 0.962·27-s − 1.67·29-s + 1.25·31-s − 0.174·33-s + 0.676·35-s − 1.31·37-s − 0.800·39-s − 1.52·43-s − 0.298·45-s − 0.437·47-s + 9/7·49-s + 0.140·51-s + 0.412·53-s − 0.134·55-s − 0.132·57-s + 1.17·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(59840\)    =    \(2^{6} \cdot 5 \cdot 11 \cdot 17\)
Sign: $1$
Analytic conductor: \(477.824\)
Root analytic conductor: \(21.8592\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 59840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.570980643\)
\(L(\frac12)\) \(\approx\) \(2.570980643\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
11 \( 1 + T \)
17 \( 1 - T \)
good3 \( 1 - T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 - 7 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 10 T + p T^{2} \)
47 \( 1 + 3 T + p T^{2} \)
53 \( 1 - 3 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 15 T + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−14.41965916763740, −13.83186043869229, −13.51143961437550, −12.89925434908867, −12.13180316792059, −11.83346041249199, −11.30782972998834, −10.75725228387080, −10.18381342083067, −9.695363434115970, −9.116794942551759, −8.533313568229130, −8.135721717390420, −7.671741799428099, −7.149564746644340, −6.467015565377097, −5.616692838485547, −5.209825240790486, −4.845641392661838, −4.094855160811968, −3.302691672911024, −2.686743978244337, −1.962677388369224, −1.733693711524981, −0.4913952344892875, 0.4913952344892875, 1.733693711524981, 1.962677388369224, 2.686743978244337, 3.302691672911024, 4.094855160811968, 4.845641392661838, 5.209825240790486, 5.616692838485547, 6.467015565377097, 7.149564746644340, 7.671741799428099, 8.135721717390420, 8.533313568229130, 9.116794942551759, 9.695363434115970, 10.18381342083067, 10.75725228387080, 11.30782972998834, 11.83346041249199, 12.13180316792059, 12.89925434908867, 13.51143961437550, 13.83186043869229, 14.41965916763740

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