↑

Properties

Label 1-53-53.52-r0-0-0
Degree 11
Conductor 5353
Sign 11
Analytic cond. 0.2461300.246130
Root an. cond. 0.2461300.246130
Motivic weight 00
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯
L(s)  = 1  − 2-s − 3-s + 4-s − 5-s + 6-s + 7-s − 8-s + 9-s + 10-s + 11-s − 12-s + 13-s − 14-s + 15-s + 16-s + 17-s − 18-s − 19-s − 20-s − 21-s − 22-s − 23-s + 24-s + 25-s − 26-s − 27-s + 28-s + ⋯

Functional equation

Λ(s)=(53s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
Λ(s)=(53s/2 ΓR(s) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 53 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 11
Conductor: 5353
Sign: 11
Analytic conductor: 0.2461300.246130
Root analytic conductor: 0.2461300.246130
Motivic weight: 00
Rational: yes
Arithmetic: yes
Character: χ53(52,⋅)\chi_{53} (52, \cdot )
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (1, 53, (0: ), 1)(1,\ 53,\ (0:\ ),\ 1)

Particular Values

L(12)L(\frac{1}{2}) ≈\approx 0.44132436980.4413243698
L(12)L(\frac12) ≈\approx 0.44132436980.4413243698
L(1)L(1) ≈\approx 0.54002494510.5400249451
L(1)L(1) ≈\approx 0.54002494510.5400249451

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad53 1 1
good2 1−T 1 - T
3 1−T 1 - T
5 1−T 1 - T
7 1+T 1 + T
11 1+T 1 + T
13 1+T 1 + T
17 1+T 1 + T
19 1−T 1 - T
23 1−T 1 - T
29 1+T 1 + T
31 1−T 1 - T
37 1+T 1 + T
41 1−T 1 - T
43 1+T 1 + T
47 1+T 1 + T
59 1+T 1 + T
61 1−T 1 - T
67 1−T 1 - T
71 1−T 1 - T
73 1−T 1 - T
79 1−T 1 - T
83 1−T 1 - T
89 1+T 1 + T
97 1+T 1 + T
show more
show less
   L(s)=∏p (1−αp p−s)−1L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−33.86152436170425418701982870885, −32.57269619264754611232473349504, −30.545620946045299761643330146871, −29.966602467693917263919139438815, −28.36648787304449427243069888533, −27.602954185546719470641618338062, −27.13416327234771392576660466512, −25.42092095490063799611210147225, −24.05560065587156226990139008675, −23.35307579450127140854381574674, −21.66216708591945991738451762447, −20.4164355781754680603999999097, −19.05859966143852274240922060246, −18.0793699649316956247127383474, −16.94426102004003420824110155822, −16.00289465879198906094236017551, −14.73731489654954081513575792093, −12.20791887663061997561285230493, −11.45107574999535622633355811796, −10.48461614494641777812922491700, −8.639675813364645526117778754596, −7.44948828207489986312977424102, −6.04706264523221344011253005798, −4.0857712719475633052890036949, −1.28711359688044475280006335894, 1.28711359688044475280006335894, 4.0857712719475633052890036949, 6.04706264523221344011253005798, 7.44948828207489986312977424102, 8.639675813364645526117778754596, 10.48461614494641777812922491700, 11.45107574999535622633355811796, 12.20791887663061997561285230493, 14.73731489654954081513575792093, 16.00289465879198906094236017551, 16.94426102004003420824110155822, 18.0793699649316956247127383474, 19.05859966143852274240922060246, 20.4164355781754680603999999097, 21.66216708591945991738451762447, 23.35307579450127140854381574674, 24.05560065587156226990139008675, 25.42092095490063799611210147225, 27.13416327234771392576660466512, 27.602954185546719470641618338062, 28.36648787304449427243069888533, 29.966602467693917263919139438815, 30.545620946045299761643330146871, 32.57269619264754611232473349504, 33.86152436170425418701982870885

Graph of the ZZ-function along the critical line