↑

Properties

Label 2-55-55.54-c4-0-8
Degree 22
Conductor 5555
Sign 11
Analytic cond. 5.685345.68534
Root an. cond. 2.384392.38439
Motivic weight 44
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3·2-s − 7·4-s + 25·5-s − 78·7-s + 69·8-s + 81·9-s − 75·10-s + 121·11-s + 162·13-s + 234·14-s − 95·16-s + 402·17-s − 243·18-s − 175·20-s − 363·22-s + 625·25-s − 486·26-s + 546·28-s − 1.59e3·31-s − 819·32-s − 1.20e3·34-s − 1.95e3·35-s − 567·36-s + 1.72e3·40-s + 3.52e3·43-s − 847·44-s + 2.02e3·45-s + ⋯
L(s)  = 1  − 3/4·2-s − 0.437·4-s + 5-s − 1.59·7-s + 1.07·8-s + 9-s − 3/4·10-s + 11-s + 0.958·13-s + 1.19·14-s − 0.371·16-s + 1.39·17-s − 3/4·18-s − 0.437·20-s − 3/4·22-s + 25-s − 0.718·26-s + 0.696·28-s − 1.66·31-s − 0.799·32-s − 1.04·34-s − 1.59·35-s − 0.437·36-s + 1.07·40-s + 1.90·43-s − 0.437·44-s + 45-s + ⋯

Functional equation

Λ(s)=(55s/2 ΓC(s) L(s)=( Λ(5−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(5-s) \end{aligned}
Λ(s)=(55s/2 ΓC(s+2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 5555    =    5⋅115 \cdot 11
Sign: 11
Analytic conductor: 5.685345.68534
Root analytic conductor: 2.384392.38439
Motivic weight: 44
Rational: yes
Arithmetic: yes
Character: χ55(54,⋅)\chi_{55} (54, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 55, ( :2), 1)(2,\ 55,\ (\ :2),\ 1)

Particular Values

L(52)L(\frac{5}{2}) ≈\approx 1.0900931561.090093156
L(12)L(\frac12) ≈\approx 1.0900931561.090093156
L(3)L(3) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad5 1−p2T 1 - p^{2} T
11 1−p2T 1 - p^{2} T
good2 1+3T+p4T2 1 + 3 T + p^{4} T^{2}
3 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
7 1+78T+p4T2 1 + 78 T + p^{4} T^{2}
13 1−162T+p4T2 1 - 162 T + p^{4} T^{2}
17 1−402T+p4T2 1 - 402 T + p^{4} T^{2}
19 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
23 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
29 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
31 1+1598T+p4T2 1 + 1598 T + p^{4} T^{2}
37 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
41 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
43 1−3522T+p4T2 1 - 3522 T + p^{4} T^{2}
47 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
53 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
59 1−3442T+p4T2 1 - 3442 T + p^{4} T^{2}
61 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
67 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
71 1+3998T+p4T2 1 + 3998 T + p^{4} T^{2}
73 1+10638T+p4T2 1 + 10638 T + p^{4} T^{2}
79 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
83 1−13602T+p4T2 1 - 13602 T + p^{4} T^{2}
89 1+15838T+p4T2 1 + 15838 T + p^{4} T^{2}
97 (1−p2T)(1+p2T) ( 1 - p^{2} T )( 1 + p^{2} T )
show more
show less
   L(s)=∏p ∏j=12(1−αj,p p−s)−1L(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

−14.37037296408660213649676044466, −13.30806682235148526042100439220, −12.60188675466443481024029637777, −10.50010570175545202432239091031, −9.656419559294173182443187840980, −9.041619573360570007835362288547, −7.16804240128046363763499258564, −5.88762702237267445877934833591, −3.77131252371471343448595855825, −1.20089647082121139804533191441, 1.20089647082121139804533191441, 3.77131252371471343448595855825, 5.88762702237267445877934833591, 7.16804240128046363763499258564, 9.041619573360570007835362288547, 9.656419559294173182443187840980, 10.50010570175545202432239091031, 12.60188675466443481024029637777, 13.30806682235148526042100439220, 14.37037296408660213649676044466

Graph of the ZZ-function along the critical line