Properties

Label 2-85680-1.1-c1-0-43
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 − 2·13-s − 17-s + 4·19-s + 2·23-s + 25-s − 2·29-s + 8·31-s − 35-s + 6·37-s + 10·41-s + 2·47-s + 49-s − 10·53-s + 2·55-s − 6·59-s − 10·61-s − 2·65-s + 8·67-s + 8·71-s + 10·73-s − 2·77-s + 6·79-s + 8·83-s − 85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.603·11-s − 0.554·13-s − 0.242·17-s + 0.917·19-s + 0.417·23-s + 1/5·25-s − 0.371·29-s + 1.43·31-s − 0.169·35-s + 0.986·37-s + 1.56·41-s + 0.291·47-s + 1/7·49-s − 1.37·53-s + 0.269·55-s − 0.781·59-s − 1.28·61-s − 0.248·65-s + 0.977·67-s + 0.949·71-s + 1.17·73-s − 0.227·77-s + 0.675·79-s + 0.878·83-s − 0.108·85-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: $\chi_{85680} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 85680,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.997894425\)
\(L(\frac12)\) \(\approx\) \(2.997894425\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 2 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 6 T + p T^{2} \)
83 \( 1 - 8 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.98821977315960, −13.52330103006902, −12.87671891611214, −12.42497623443425, −12.09500622410276, −11.32990089690847, −11.04429477069453, −10.43243763959288, −9.710662639651130, −9.417357628614294, −9.223109846378599, −8.282242198714561, −7.880747715604386, −7.263079399809107, −6.732154442997730, −6.179491218677025, −5.814360597314551, −4.995243015291568, −4.613876691492710, −3.931235069194626, −3.205182635680523, −2.702224906856452, −2.065606116553847, −1.195550858888380, −0.6103660344052941, 0.6103660344052941, 1.195550858888380, 2.065606116553847, 2.702224906856452, 3.205182635680523, 3.931235069194626, 4.613876691492710, 4.995243015291568, 5.814360597314551, 6.179491218677025, 6.732154442997730, 7.263079399809107, 7.880747715604386, 8.282242198714561, 9.223109846378599, 9.417357628614294, 9.710662639651130, 10.43243763959288, 11.04429477069453, 11.32990089690847, 12.09500622410276, 12.42497623443425, 12.87671891611214, 13.52330103006902, 13.98821977315960

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