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Properties

Label 2-115920-1.1-c1-0-128
Degree $2$
Conductor $115920$
Sign $-1$
Analytic cond. $925.625$
Root an. cond. $30.4240$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 2·11-s + 6·13-s − 2·19-s + 23-s + 25-s + 4·29-s − 10·31-s + 35-s + 10·37-s + 6·41-s − 6·43-s + 10·47-s + 49-s − 10·53-s − 2·55-s − 4·59-s + 6·61-s + 6·65-s + 2·67-s − 2·71-s − 10·73-s − 2·77-s − 16·79-s + 6·83-s + 6·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s + 1.66·13-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 0.742·29-s − 1.79·31-s + 0.169·35-s + 1.64·37-s + 0.937·41-s − 0.914·43-s + 1.45·47-s + 1/7·49-s − 1.37·53-s − 0.269·55-s − 0.520·59-s + 0.768·61-s + 0.744·65-s + 0.244·67-s − 0.237·71-s − 1.17·73-s − 0.227·77-s − 1.80·79-s + 0.658·83-s + 0.635·89-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 115920 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(115920\)    =    \(2^{4} \cdot 3^{2} \cdot 5 \cdot 7 \cdot 23\)
Sign: $-1$
Analytic conductor: \(925.625\)
Root analytic conductor: \(30.4240\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
23 \( 1 - T \)
good11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 16 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.80610096229130, −13.26474974089104, −12.93274282769170, −12.64145626566403, −11.77022706856731, −11.35757118771769, −10.86897649933900, −10.56719222065563, −10.04765580731906, −9.244196201593625, −9.033503260138608, −8.428425156478605, −7.944262044043800, −7.471894895651591, −6.811750935911235, −6.160992733184140, −5.889643152262068, −5.311437593926974, −4.673707466535370, −4.089647380845257, −3.568832436223087, −2.816345784600445, −2.303910204213944, −1.488368111917740, −1.049025852671556, 0, 1.049025852671556, 1.488368111917740, 2.303910204213944, 2.816345784600445, 3.568832436223087, 4.089647380845257, 4.673707466535370, 5.311437593926974, 5.889643152262068, 6.160992733184140, 6.811750935911235, 7.471894895651591, 7.944262044043800, 8.428425156478605, 9.033503260138608, 9.244196201593625, 10.04765580731906, 10.56719222065563, 10.86897649933900, 11.35757118771769, 11.77022706856731, 12.64145626566403, 12.93274282769170, 13.26474974089104, 13.80610096229130

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