Properties

Label 2-115920-1.1-c1-0-11
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 + 2·13-s − 6·17-s + 4·19-s − 23-s + 25-s − 6·29-s + 4·31-s − 35-s + 2·37-s + 6·41-s − 8·43-s + 12·47-s + 49-s + 6·53-s − 12·59-s + 2·61-s + 2·65-s − 8·67-s + 2·73-s − 8·79-s + 12·83-s − 6·85-s − 6·89-s − 2·91-s + 4·95-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.554·13-s − 1.45·17-s + 0.917·19-s − 0.208·23-s + 1/5·25-s − 1.11·29-s + 0.718·31-s − 0.169·35-s + 0.328·37-s + 0.937·41-s − 1.21·43-s + 1.75·47-s + 1/7·49-s + 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.248·65-s − 0.977·67-s + 0.234·73-s − 0.900·79-s + 1.31·83-s − 0.650·85-s − 0.635·89-s − 0.209·91-s + 0.410·95-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

\(L(1)\) \(\approx\) \(2.105399131\)
\(L(\frac12)\) \(\approx\) \(2.105399131\)
\(L(\frac{3}{2})\) 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 \( 1 + 6 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
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

−13.56869280830700, −13.30182937198095, −12.68243581963641, −12.15617664289424, −11.68577907563822, −11.01770527429545, −10.84521646139164, −10.13392654895327, −9.659079652556586, −9.186559991663752, −8.784478284405266, −8.275234361439006, −7.461197798246447, −7.240214057229371, −6.432629822340072, −6.148762833096527, −5.575883681603004, −5.010205729025273, −4.291337916865864, −3.909751635341366, −3.124062581575114, −2.603911080808049, −1.958282904610860, −1.269951854769936, −0.4545681757956093, 0.4545681757956093, 1.269951854769936, 1.958282904610860, 2.603911080808049, 3.124062581575114, 3.909751635341366, 4.291337916865864, 5.010205729025273, 5.575883681603004, 6.148762833096527, 6.432629822340072, 7.240214057229371, 7.461197798246447, 8.275234361439006, 8.784478284405266, 9.186559991663752, 9.659079652556586, 10.13392654895327, 10.84521646139164, 11.01770527429545, 11.68577907563822, 12.15617664289424, 12.68243581963641, 13.30182937198095, 13.56869280830700

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