↑

Properties

Label 2-153-1.1-c3-0-16
Degree 22
Conductor 153153
Sign −1-1
Analytic cond. 9.027299.02729
Root an. cond. 3.004543.00454
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.47·2-s − 5.81·4-s + 11.3·5-s − 28.3·7-s − 20.4·8-s + 16.7·10-s − 56.2·11-s + 24.3·13-s − 41.9·14-s + 16.4·16-s + 17·17-s − 34.0·19-s − 65.9·20-s − 83.0·22-s − 108.·23-s + 3.55·25-s + 36.0·26-s + 165.·28-s − 266.·29-s − 207.·31-s + 187.·32-s + 25.1·34-s − 321.·35-s + 380.·37-s − 50.2·38-s − 231.·40-s + 451.·41-s + ⋯
L(s)  = 1  + 0.522·2-s − 0.727·4-s + 1.01·5-s − 1.53·7-s − 0.901·8-s + 0.529·10-s − 1.54·11-s + 0.520·13-s − 0.800·14-s + 0.256·16-s + 0.242·17-s − 0.410·19-s − 0.737·20-s − 0.804·22-s − 0.982·23-s + 0.0284·25-s + 0.271·26-s + 1.11·28-s − 1.70·29-s − 1.20·31-s + 1.03·32-s + 0.126·34-s − 1.55·35-s + 1.68·37-s − 0.214·38-s − 0.914·40-s + 1.72·41-s + ⋯

Functional equation

Λ(s)=(153s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(153s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 153153    =    32⋅173^{2} \cdot 17
Sign: −1-1
Analytic conductor: 9.027299.02729
Root analytic conductor: 3.004543.00454
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 153, ( :3/2), −1)(2,\ 153,\ (\ :3/2),\ -1)

Particular Values

L(2)L(2) == 00
L(12)L(\frac12) == 00
L(52)L(\frac{5}{2}) 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)
bad3 1 1
17 1−17T 1 - 17T
good2 1−1.47T+8T2 1 - 1.47T + 8T^{2}
5 1−11.3T+125T2 1 - 11.3T + 125T^{2}
7 1+28.3T+343T2 1 + 28.3T + 343T^{2}
11 1+56.2T+1.33e3T2 1 + 56.2T + 1.33e3T^{2}
13 1−24.3T+2.19e3T2 1 - 24.3T + 2.19e3T^{2}
19 1+34.0T+6.85e3T2 1 + 34.0T + 6.85e3T^{2}
23 1+108.T+1.21e4T2 1 + 108.T + 1.21e4T^{2}
29 1+266.T+2.43e4T2 1 + 266.T + 2.43e4T^{2}
31 1+207.T+2.97e4T2 1 + 207.T + 2.97e4T^{2}
37 1−380.T+5.06e4T2 1 - 380.T + 5.06e4T^{2}
41 1−451.T+6.89e4T2 1 - 451.T + 6.89e4T^{2}
43 1−395.T+7.95e4T2 1 - 395.T + 7.95e4T^{2}
47 1+179.T+1.03e5T2 1 + 179.T + 1.03e5T^{2}
53 1+184.T+1.48e5T2 1 + 184.T + 1.48e5T^{2}
59 1+151.T+2.05e5T2 1 + 151.T + 2.05e5T^{2}
61 1−59.0T+2.26e5T2 1 - 59.0T + 2.26e5T^{2}
67 1+56.7T+3.00e5T2 1 + 56.7T + 3.00e5T^{2}
71 1−265.T+3.57e5T2 1 - 265.T + 3.57e5T^{2}
73 1+704.T+3.89e5T2 1 + 704.T + 3.89e5T^{2}
79 1−509.T+4.93e5T2 1 - 509.T + 4.93e5T^{2}
83 1−122.T+5.71e5T2 1 - 122.T + 5.71e5T^{2}
89 1+710.T+7.04e5T2 1 + 710.T + 7.04e5T^{2}
97 1+834.T+9.12e5T2 1 + 834.T + 9.12e5T^{2}
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

−12.70150392833923993168985048737, −10.84234923602669132307129689891, −9.742029771866160122736792023100, −9.284618405104493347295221023592, −7.77153077594818912779379293826, −6.04687634561477708950953865143, −5.62869598682938290675957681616, −3.93722421679993483925939503786, −2.61568117400199109494512535237, 0, 2.61568117400199109494512535237, 3.93722421679993483925939503786, 5.62869598682938290675957681616, 6.04687634561477708950953865143, 7.77153077594818912779379293826, 9.284618405104493347295221023592, 9.742029771866160122736792023100, 10.84234923602669132307129689891, 12.70150392833923993168985048737

Graph of the ZZ-function along the critical line