Properties

Label 2-123840-1.1-c1-0-61
Degree $2$
Conductor $123840$
Sign $1$
Analytic cond. $988.867$
Root an. cond. $31.4462$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 5-s + 2·7-s + 2·11-s + 2·13-s + 4·17-s − 6·19-s + 6·23-s + 25-s − 10·29-s + 8·31-s + 2·35-s − 2·37-s − 2·41-s − 43-s − 2·47-s − 3·49-s + 10·53-s + 2·55-s − 2·59-s + 12·61-s + 2·65-s − 12·67-s + 16·71-s + 16·73-s + 4·77-s + 8·79-s + 12·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.755·7-s + 0.603·11-s + 0.554·13-s + 0.970·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s − 1.85·29-s + 1.43·31-s + 0.338·35-s − 0.328·37-s − 0.312·41-s − 0.152·43-s − 0.291·47-s − 3/7·49-s + 1.37·53-s + 0.269·55-s − 0.260·59-s + 1.53·61-s + 0.248·65-s − 1.46·67-s + 1.89·71-s + 1.87·73-s + 0.455·77-s + 0.900·79-s + 1.31·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 123840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(123840\)    =    \(2^{6} \cdot 3^{2} \cdot 5 \cdot 43\)
Sign: $1$
Analytic conductor: \(988.867\)
Root analytic conductor: \(31.4462\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{123840} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 123840,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.980697366\)
\(L(\frac12)\) \(\approx\) \(3.980697366\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
43 \( 1 + T \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + 10 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
47 \( 1 + 2 T + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 12 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 16 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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

−13.45994372226780, −13.22098521728065, −12.46516952998044, −12.21415668220293, −11.43387735921668, −11.19730187670142, −10.66229273682474, −10.20661931232345, −9.567120093892552, −9.156108032395344, −8.629884112647921, −8.121627889363340, −7.768672741713220, −6.888330918046123, −6.648373646987520, −6.041768405365033, −5.352631186924323, −5.091528746299688, −4.340766226704009, −3.776463869790034, −3.298965147930299, −2.417406602415846, −1.905401834848984, −1.263110041300922, −0.6451280424215831, 0.6451280424215831, 1.263110041300922, 1.905401834848984, 2.417406602415846, 3.298965147930299, 3.776463869790034, 4.340766226704009, 5.091528746299688, 5.352631186924323, 6.041768405365033, 6.648373646987520, 6.888330918046123, 7.768672741713220, 8.121627889363340, 8.629884112647921, 9.156108032395344, 9.567120093892552, 10.20661931232345, 10.66229273682474, 11.19730187670142, 11.43387735921668, 12.21415668220293, 12.46516952998044, 13.22098521728065, 13.45994372226780

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