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Properties

Label 2-115920-1.1-c1-0-52
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 + 6·11-s − 2·13-s + 4·17-s − 2·19-s + 23-s + 25-s + 8·29-s − 6·31-s − 35-s + 2·37-s + 10·41-s + 2·43-s − 2·47-s + 49-s + 6·53-s + 6·55-s + 4·59-s + 6·61-s − 2·65-s + 2·67-s + 10·71-s − 10·73-s − 6·77-s − 8·79-s + 14·83-s + ⋯
L(s)  = 1  + 0.447·5-s − 0.377·7-s + 1.80·11-s − 0.554·13-s + 0.970·17-s − 0.458·19-s + 0.208·23-s + 1/5·25-s + 1.48·29-s − 1.07·31-s − 0.169·35-s + 0.328·37-s + 1.56·41-s + 0.304·43-s − 0.291·47-s + 1/7·49-s + 0.824·53-s + 0.809·55-s + 0.520·59-s + 0.768·61-s − 0.248·65-s + 0.244·67-s + 1.18·71-s − 1.17·73-s − 0.683·77-s − 0.900·79-s + 1.53·83-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 115920,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.718835328\)
\(L(\frac12)\) \(\approx\) \(3.718835328\)
\(L(\frac{3}{2})\) 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 \( 1 - 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 8 T + p T^{2} \)
31 \( 1 + 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 6 T + p T^{2} \)
67 \( 1 - 2 T + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
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

−13.67208150122052, −13.05522925477477, −12.55784676599354, −12.21624212103682, −11.74797379045404, −11.21185307188801, −10.66339978000037, −10.05420286633826, −9.696609260435811, −9.226287470623803, −8.810466230660478, −8.257028967727903, −7.533438055368466, −7.075727999375547, −6.520983614202428, −6.171830850904786, −5.557282036832442, −5.019878940364192, −4.211443978990439, −3.935991448835139, −3.198913502050436, −2.593082630341583, −1.926214043104472, −1.158202738339974, −0.6658941190458171, 0.6658941190458171, 1.158202738339974, 1.926214043104472, 2.593082630341583, 3.198913502050436, 3.935991448835139, 4.211443978990439, 5.019878940364192, 5.557282036832442, 6.171830850904786, 6.520983614202428, 7.075727999375547, 7.533438055368466, 8.257028967727903, 8.810466230660478, 9.226287470623803, 9.696609260435811, 10.05420286633826, 10.66339978000037, 11.21185307188801, 11.74797379045404, 12.21624212103682, 12.55784676599354, 13.05522925477477, 13.67208150122052

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