Properties

Label 2-115920-1.1-c1-0-13
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 5-s − 7-s − 3·11-s + 3·13-s − 3·17-s + 23-s + 25-s + 9·29-s − 2·31-s − 35-s − 4·37-s − 4·41-s + 2·43-s + 13·47-s + 49-s + 2·53-s − 3·55-s − 6·59-s − 6·61-s + 3·65-s + 12·67-s − 8·71-s − 6·73-s + 3·77-s + 13·79-s + 12·83-s − 3·85-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s − 0.904·11-s + 0.832·13-s − 0.727·17-s + 0.208·23-s + 1/5·25-s + 1.67·29-s − 0.359·31-s − 0.169·35-s − 0.657·37-s − 0.624·41-s + 0.304·43-s + 1.89·47-s + 1/7·49-s + 0.274·53-s − 0.404·55-s − 0.781·59-s − 0.768·61-s + 0.372·65-s + 1.46·67-s − 0.949·71-s − 0.702·73-s + 0.341·77-s + 1.46·79-s + 1.31·83-s − 0.325·85-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: $\chi_{115920} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.179282197\)
\(L(\frac12)\) \(\approx\) \(2.179282197\)
\(L(\frac{3}{2})\) 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 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 - 13 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 13 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.46184764445539, −13.27573532798922, −12.73268830141195, −12.04210190269012, −11.90202970358865, −10.86715627299210, −10.67945613258287, −10.40856883313414, −9.634143237426775, −9.188115169483613, −8.690876753655632, −8.263567199720459, −7.663119654829254, −7.039276728219622, −6.549543761895103, −6.113189327512672, −5.505666552664774, −5.030175620909865, −4.411834709883488, −3.803384174745414, −3.113353457681651, −2.606708149776291, −2.022858768408325, −1.209353940099876, −0.4737205530373174, 0.4737205530373174, 1.209353940099876, 2.022858768408325, 2.606708149776291, 3.113353457681651, 3.803384174745414, 4.411834709883488, 5.030175620909865, 5.505666552664774, 6.113189327512672, 6.549543761895103, 7.039276728219622, 7.663119654829254, 8.263567199720459, 8.690876753655632, 9.188115169483613, 9.634143237426775, 10.40856883313414, 10.67945613258287, 10.86715627299210, 11.90202970358865, 12.04210190269012, 12.73268830141195, 13.27573532798922, 13.46184764445539

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