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Properties

Label 2-85680-1.1-c1-0-10
Degree $2$
Conductor $85680$
Sign $1$
Analytic cond. $684.158$
Root an. cond. $26.1564$
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 − 7-s + 2·11-s − 5·13-s − 17-s − 2·19-s − 23-s + 25-s − 8·29-s − 31-s − 35-s − 3·37-s + 7·41-s − 47-s + 49-s + 8·53-s + 2·55-s − 7·61-s − 5·65-s − 16·67-s − 10·71-s − 8·73-s − 2·77-s + 6·79-s − 13·83-s − 85-s + 2·89-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s − 1.38·13-s − 0.242·17-s − 0.458·19-s − 0.208·23-s + 1/5·25-s − 1.48·29-s − 0.179·31-s − 0.169·35-s − 0.493·37-s + 1.09·41-s − 0.145·47-s + 1/7·49-s + 1.09·53-s + 0.269·55-s − 0.896·61-s − 0.620·65-s − 1.95·67-s − 1.18·71-s − 0.936·73-s − 0.227·77-s + 0.675·79-s − 1.42·83-s − 0.108·85-s + 0.211·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 85680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(85680\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 17\)
Sign: $1$
Analytic conductor: \(684.158\)
Root analytic conductor: \(26.1564\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.018036465\)
\(L(\frac12)\) \(\approx\) \(1.018036465\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 - T \)
7 \( 1 + T \)
17 \( 1 + T \)
good11 \( 1 - 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 + T + p T^{2} \)
53 \( 1 - 8 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 10 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 + 13 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.90091415022155, −13.40142095406797, −12.91827841555690, −12.51017392329230, −11.88368954283755, −11.64631162549146, −10.73889398294991, −10.53746553765573, −9.816120686432512, −9.433474479112483, −9.030627892761291, −8.510210856689637, −7.666441153126391, −7.291272790310786, −6.856166290240531, −6.121507750631363, −5.744616684688011, −5.160569309869962, −4.382294674213399, −4.091843640911220, −3.203431199071578, −2.656058952533733, −1.996084150972520, −1.423523789520071, −0.3115024213168838, 0.3115024213168838, 1.423523789520071, 1.996084150972520, 2.656058952533733, 3.203431199071578, 4.091843640911220, 4.382294674213399, 5.160569309869962, 5.744616684688011, 6.121507750631363, 6.856166290240531, 7.291272790310786, 7.666441153126391, 8.510210856689637, 9.030627892761291, 9.433474479112483, 9.816120686432512, 10.53746553765573, 10.73889398294991, 11.64631162549146, 11.88368954283755, 12.51017392329230, 12.91827841555690, 13.40142095406797, 13.90091415022155

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