Properties

Label 2-115920-1.1-c1-0-127
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 + 6·17-s − 23-s + 25-s + 10·29-s − 8·31-s + 35-s − 6·37-s + 6·41-s + 8·47-s + 49-s + 14·53-s + 4·59-s − 14·61-s − 2·65-s − 16·67-s − 8·71-s − 6·73-s + 6·85-s + 14·89-s − 2·91-s + 2·97-s + 101-s + 103-s + 107-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.554·13-s + 1.45·17-s − 0.208·23-s + 1/5·25-s + 1.85·29-s − 1.43·31-s + 0.169·35-s − 0.986·37-s + 0.937·41-s + 1.16·47-s + 1/7·49-s + 1.92·53-s + 0.520·59-s − 1.79·61-s − 0.248·65-s − 1.95·67-s − 0.949·71-s − 0.702·73-s + 0.650·85-s + 1.48·89-s − 0.209·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-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 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 14 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
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

−13.76192798531835, −13.56415050743485, −12.78326196502938, −12.31391181692414, −11.97500383417856, −11.60919382784761, −10.69602203301174, −10.36145981422068, −10.17878304200834, −9.376992190151071, −8.914693959221073, −8.578601387213138, −7.680027932499163, −7.513881593374227, −6.995021182117281, −6.086783835251668, −5.908191976301997, −5.155018241828905, −4.852456229941591, −4.070592229210582, −3.539956050333846, −2.766498642794259, −2.377128920823502, −1.471360455139734, −1.041296958217104, 0, 1.041296958217104, 1.471360455139734, 2.377128920823502, 2.766498642794259, 3.539956050333846, 4.070592229210582, 4.852456229941591, 5.155018241828905, 5.908191976301997, 6.086783835251668, 6.995021182117281, 7.513881593374227, 7.680027932499163, 8.578601387213138, 8.914693959221073, 9.376992190151071, 10.17878304200834, 10.36145981422068, 10.69602203301174, 11.60919382784761, 11.97500383417856, 12.31391181692414, 12.78326196502938, 13.56415050743485, 13.76192798531835

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