Properties

Label 2-115920-1.1-c1-0-120
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 − 2·17-s + 6·19-s + 23-s + 25-s + 2·29-s − 4·31-s − 35-s − 12·37-s + 10·41-s + 12·43-s + 4·47-s + 49-s + 2·53-s − 6·59-s − 2·61-s + 2·65-s + 4·67-s − 6·71-s + 2·73-s − 14·79-s − 12·83-s − 2·85-s − 2·89-s − 2·91-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 0.554·13-s − 0.485·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.169·35-s − 1.97·37-s + 1.56·41-s + 1.82·43-s + 0.583·47-s + 1/7·49-s + 0.274·53-s − 0.781·59-s − 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.712·71-s + 0.234·73-s − 1.57·79-s − 1.31·83-s − 0.216·85-s − 0.211·89-s − 0.209·91-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 \( 1 + 2 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 12 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 6 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 14 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 2 T + p T^{2} \)
97 \( 1 - 4 T + p T^{2} \)
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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

−13.79591579721935, −13.50123367883149, −12.71036834241029, −12.59431806224619, −11.96241813055239, −11.24578935393932, −11.07583368790963, −10.26156170285887, −10.11835900553990, −9.280210749098028, −8.978428136137406, −8.692974044424893, −7.685564372982509, −7.468966084447841, −6.860043402286126, −6.297249247075784, −5.697039898883053, −5.430987794183968, −4.680170161926507, −4.049427936445424, −3.513857773337609, −2.851938775015769, −2.370531990392972, −1.484985415591954, −0.9808222613084741, 0, 0.9808222613084741, 1.484985415591954, 2.370531990392972, 2.851938775015769, 3.513857773337609, 4.049427936445424, 4.680170161926507, 5.430987794183968, 5.697039898883053, 6.297249247075784, 6.860043402286126, 7.468966084447841, 7.685564372982509, 8.692974044424893, 8.978428136137406, 9.280210749098028, 10.11835900553990, 10.26156170285887, 11.07583368790963, 11.24578935393932, 11.96241813055239, 12.59431806224619, 12.71036834241029, 13.50123367883149, 13.79591579721935

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