Properties

Label 2-54080-1.1-c1-0-62
Degree $2$
Conductor $54080$
Sign $1$
Analytic cond. $431.830$
Root an. cond. $20.7805$
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  + 5-s + 4·7-s − 3·9-s + 4·11-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 2·29-s + 8·31-s + 4·35-s + 6·37-s + 6·41-s + 8·43-s − 3·45-s − 4·47-s + 9·49-s − 6·53-s + 4·55-s − 4·59-s + 2·61-s − 12·63-s + 8·67-s + 6·73-s + 16·77-s + 9·81-s − 16·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 1.51·7-s − 9-s + 1.20·11-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.371·29-s + 1.43·31-s + 0.676·35-s + 0.986·37-s + 0.937·41-s + 1.21·43-s − 0.447·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.539·55-s − 0.520·59-s + 0.256·61-s − 1.51·63-s + 0.977·67-s + 0.702·73-s + 1.82·77-s + 81-s − 1.75·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 54080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(54080\)    =    \(2^{6} \cdot 5 \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(431.830\)
Root analytic conductor: \(20.7805\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 54080,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(4.657875795\)
\(L(\frac12)\) \(\approx\) \(4.657875795\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
13 \( 1 \)
good3 \( 1 + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−14.42348078809323, −14.04548033972279, −13.72076622065296, −12.82768086838973, −12.34124139020279, −11.69324890728209, −11.27765787259393, −11.20640274989453, −10.29260906753295, −9.770240192133980, −9.051151164743510, −8.887744336334850, −8.003160391213557, −7.856564914211415, −7.072370837446894, −6.334477625705576, −5.936247019670364, −5.264687534250796, −4.775611626839977, −4.248721042309023, −3.387502791587691, −2.757864233261148, −2.107630389840526, −1.158413653218510, −0.9201653962135482, 0.9201653962135482, 1.158413653218510, 2.107630389840526, 2.757864233261148, 3.387502791587691, 4.248721042309023, 4.775611626839977, 5.264687534250796, 5.936247019670364, 6.334477625705576, 7.072370837446894, 7.856564914211415, 8.003160391213557, 8.887744336334850, 9.051151164743510, 9.770240192133980, 10.29260906753295, 11.20640274989453, 11.27765787259393, 11.69324890728209, 12.34124139020279, 12.82768086838973, 13.72076622065296, 14.04548033972279, 14.42348078809323

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