Properties

Label 2-115920-1.1-c1-0-106
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 − 2·13-s − 23-s + 25-s − 8·29-s + 4·31-s + 35-s + 6·37-s + 12·43-s − 4·47-s + 49-s − 4·53-s + 4·59-s + 10·61-s − 2·65-s − 4·67-s − 8·71-s + 12·83-s − 10·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.554·13-s − 0.208·23-s + 1/5·25-s − 1.48·29-s + 0.718·31-s + 0.169·35-s + 0.986·37-s + 1.82·43-s − 0.583·47-s + 1/7·49-s − 0.549·53-s + 0.520·59-s + 1.28·61-s − 0.248·65-s − 0.488·67-s − 0.949·71-s + 1.31·83-s − 1.05·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-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: \(1\)
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 \( 1 + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 4 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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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

−13.83345950765770, −13.32856968897098, −12.94364976014820, −12.40394535619497, −11.93044750353012, −11.36525082615404, −10.97155305056917, −10.46536773758919, −9.838255449854144, −9.492613626403084, −9.043571267826845, −8.395759951962088, −7.843281055660675, −7.480920244902655, −6.855469842932193, −6.307152341779278, −5.688566214263385, −5.359333656154273, −4.639677283710753, −4.175871104078990, −3.566637503662640, −2.700379157469089, −2.354576177272305, −1.608004536871827, −0.9339818619826122, 0, 0.9339818619826122, 1.608004536871827, 2.354576177272305, 2.700379157469089, 3.566637503662640, 4.175871104078990, 4.639677283710753, 5.359333656154273, 5.688566214263385, 6.307152341779278, 6.855469842932193, 7.480920244902655, 7.843281055660675, 8.395759951962088, 9.043571267826845, 9.492613626403084, 9.838255449854144, 10.46536773758919, 10.97155305056917, 11.36525082615404, 11.93044750353012, 12.40394535619497, 12.94364976014820, 13.32856968897098, 13.83345950765770

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