↑

Properties

Label 2-1575-1.1-c3-0-59
Degree 22
Conductor 15751575
Sign 11
Analytic cond. 92.928092.9280
Root an. cond. 9.639919.63991
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 3.53·2-s + 4.46·4-s + 7·7-s + 12.4·8-s + 2.93·11-s + 19.0·13-s − 24.7·14-s − 79.7·16-s + 122.·17-s + 107.·19-s − 10.3·22-s + 210.·23-s − 67.3·26-s + 31.2·28-s − 95.4·29-s − 94.3·31-s + 181.·32-s − 432.·34-s − 97.1·37-s − 379.·38-s + 491.·41-s + 43.0·43-s + 13.1·44-s − 743.·46-s + 473.·47-s + 49·49-s + 85.1·52-s + ⋯
L(s)  = 1  − 1.24·2-s + 0.558·4-s + 0.377·7-s + 0.551·8-s + 0.0805·11-s + 0.406·13-s − 0.471·14-s − 1.24·16-s + 1.74·17-s + 1.29·19-s − 0.100·22-s + 1.90·23-s − 0.507·26-s + 0.211·28-s − 0.611·29-s − 0.546·31-s + 1.00·32-s − 2.18·34-s − 0.431·37-s − 1.61·38-s + 1.87·41-s + 0.152·43-s + 0.0449·44-s − 2.38·46-s + 1.46·47-s + 0.142·49-s + 0.227·52-s + ⋯

Functional equation

Λ(s)=(1575s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(1575s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 15751575    =    32⋅52⋅73^{2} \cdot 5^{2} \cdot 7
Sign: 11
Analytic conductor: 92.928092.9280
Root analytic conductor: 9.639919.63991
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1575, ( :3/2), 1)(2,\ 1575,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.4378577691.437857769
L(12)L(\frac12) ≈\approx 1.4378577691.437857769
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
5 1 1
7 1−7T 1 - 7T
good2 1+3.53T+8T2 1 + 3.53T + 8T^{2}
11 1−2.93T+1.33e3T2 1 - 2.93T + 1.33e3T^{2}
13 1−19.0T+2.19e3T2 1 - 19.0T + 2.19e3T^{2}
17 1−122.T+4.91e3T2 1 - 122.T + 4.91e3T^{2}
19 1−107.T+6.85e3T2 1 - 107.T + 6.85e3T^{2}
23 1−210.T+1.21e4T2 1 - 210.T + 1.21e4T^{2}
29 1+95.4T+2.43e4T2 1 + 95.4T + 2.43e4T^{2}
31 1+94.3T+2.97e4T2 1 + 94.3T + 2.97e4T^{2}
37 1+97.1T+5.06e4T2 1 + 97.1T + 5.06e4T^{2}
41 1−491.T+6.89e4T2 1 - 491.T + 6.89e4T^{2}
43 1−43.0T+7.95e4T2 1 - 43.0T + 7.95e4T^{2}
47 1−473.T+1.03e5T2 1 - 473.T + 1.03e5T^{2}
53 1+183.T+1.48e5T2 1 + 183.T + 1.48e5T^{2}
59 1−760.T+2.05e5T2 1 - 760.T + 2.05e5T^{2}
61 1+198.T+2.26e5T2 1 + 198.T + 2.26e5T^{2}
67 1−309.T+3.00e5T2 1 - 309.T + 3.00e5T^{2}
71 1+665.T+3.57e5T2 1 + 665.T + 3.57e5T^{2}
73 1+621.T+3.89e5T2 1 + 621.T + 3.89e5T^{2}
79 1+24.7T+4.93e5T2 1 + 24.7T + 4.93e5T^{2}
83 1+406.T+5.71e5T2 1 + 406.T + 5.71e5T^{2}
89 1+261.T+7.04e5T2 1 + 261.T + 7.04e5T^{2}
97 1−1.00e3T+9.12e5T2 1 - 1.00e3T + 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

−9.153353705586358740942453497971, −8.368641941146238382480236027923, −7.43914919027476010469579570411, −7.24818549978646837826075463789, −5.76931935054066266328814399972, −5.08914932039572729156213172901, −3.88725987228588153584368358120, −2.81147344313458567046704702605, −1.37364563684643671998184723910, −0.825756224556272554430617877710, 0.825756224556272554430617877710, 1.37364563684643671998184723910, 2.81147344313458567046704702605, 3.88725987228588153584368358120, 5.08914932039572729156213172901, 5.76931935054066266328814399972, 7.24818549978646837826075463789, 7.43914919027476010469579570411, 8.368641941146238382480236027923, 9.153353705586358740942453497971

Graph of the ZZ-function along the critical line