Properties

Label 2-115920-1.1-c1-0-122
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 7-s − 2·11-s − 5·13-s + 6·17-s + 6·19-s + 23-s + 25-s + 3·29-s + 3·31-s + 35-s + 7·41-s + 6·43-s − 9·47-s + 49-s − 2·55-s − 4·59-s − 4·61-s − 5·65-s − 8·67-s − 13·71-s − 11·73-s − 2·77-s − 6·79-s + 6·83-s + 6·85-s − 14·89-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.377·7-s − 0.603·11-s − 1.38·13-s + 1.45·17-s + 1.37·19-s + 0.208·23-s + 1/5·25-s + 0.557·29-s + 0.538·31-s + 0.169·35-s + 1.09·41-s + 0.914·43-s − 1.31·47-s + 1/7·49-s − 0.269·55-s − 0.520·59-s − 0.512·61-s − 0.620·65-s − 0.977·67-s − 1.54·71-s − 1.28·73-s − 0.227·77-s − 0.675·79-s + 0.658·83-s + 0.650·85-s − 1.48·89-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 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 3 T + p T^{2} \)
31 \( 1 - 3 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 - 7 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + 13 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 + 6 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.00716175223358, −13.29596428931510, −12.94452938628505, −12.23101206332925, −12.03234919345748, −11.54392044870451, −10.86459740053307, −10.25733177543055, −10.04497832406512, −9.458451149124393, −9.126348070130065, −8.292219627641968, −7.810702901979450, −7.447192971682320, −7.057784125693992, −6.210860401231249, −5.607687808385696, −5.372852673951858, −4.659777181835532, −4.351523919581026, −3.190996248800823, −2.981373665926056, −2.374232524349245, −1.483529430657002, −1.001268566281686, 0, 1.001268566281686, 1.483529430657002, 2.374232524349245, 2.981373665926056, 3.190996248800823, 4.351523919581026, 4.659777181835532, 5.372852673951858, 5.607687808385696, 6.210860401231249, 7.057784125693992, 7.447192971682320, 7.810702901979450, 8.292219627641968, 9.126348070130065, 9.458451149124393, 10.04497832406512, 10.25733177543055, 10.86459740053307, 11.54392044870451, 12.03234919345748, 12.23101206332925, 12.94452938628505, 13.29596428931510, 14.00716175223358

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