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Properties

Label 2-315-1.1-c3-0-21
Degree 22
Conductor 315315
Sign −1-1
Analytic cond. 18.585618.5856
Root an. cond. 4.311104.31110
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.229·2-s − 7.94·4-s − 5·5-s + 7·7-s + 3.66·8-s + 1.14·10-s + 51.0·11-s + 46.0·13-s − 1.60·14-s + 62.7·16-s − 72.7·17-s − 123.·19-s + 39.7·20-s − 11.7·22-s − 156.·23-s + 25·25-s − 10.5·26-s − 55.6·28-s − 191.·29-s − 116.·31-s − 43.7·32-s + 16.7·34-s − 35·35-s + 83.1·37-s + 28.4·38-s − 18.3·40-s − 466.·41-s + ⋯
L(s)  = 1  − 0.0812·2-s − 0.993·4-s − 0.447·5-s + 0.377·7-s + 0.161·8-s + 0.0363·10-s + 1.39·11-s + 0.983·13-s − 0.0307·14-s + 0.980·16-s − 1.03·17-s − 1.49·19-s + 0.444·20-s − 0.113·22-s − 1.41·23-s + 0.200·25-s − 0.0798·26-s − 0.375·28-s − 1.22·29-s − 0.673·31-s − 0.241·32-s + 0.0843·34-s − 0.169·35-s + 0.369·37-s + 0.121·38-s − 0.0724·40-s − 1.77·41-s + ⋯

Functional equation

Λ(s)=(315s/2 ΓC(s) L(s)=(− Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}
Λ(s)=(315s/2 ΓC(s+3/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 315 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 315315    =    32⋅5⋅73^{2} \cdot 5 \cdot 7
Sign: −1-1
Analytic conductor: 18.585618.5856
Root analytic conductor: 4.311104.31110
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 315, ( :3/2), −1)(2,\ 315,\ (\ :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
5 1+5T 1 + 5T
7 1−7T 1 - 7T
good2 1+0.229T+8T2 1 + 0.229T + 8T^{2}
11 1−51.0T+1.33e3T2 1 - 51.0T + 1.33e3T^{2}
13 1−46.0T+2.19e3T2 1 - 46.0T + 2.19e3T^{2}
17 1+72.7T+4.91e3T2 1 + 72.7T + 4.91e3T^{2}
19 1+123.T+6.85e3T2 1 + 123.T + 6.85e3T^{2}
23 1+156.T+1.21e4T2 1 + 156.T + 1.21e4T^{2}
29 1+191.T+2.43e4T2 1 + 191.T + 2.43e4T^{2}
31 1+116.T+2.97e4T2 1 + 116.T + 2.97e4T^{2}
37 1−83.1T+5.06e4T2 1 - 83.1T + 5.06e4T^{2}
41 1+466.T+6.89e4T2 1 + 466.T + 6.89e4T^{2}
43 1−422.T+7.95e4T2 1 - 422.T + 7.95e4T^{2}
47 1+268.T+1.03e5T2 1 + 268.T + 1.03e5T^{2}
53 1+310.T+1.48e5T2 1 + 310.T + 1.48e5T^{2}
59 1−709.T+2.05e5T2 1 - 709.T + 2.05e5T^{2}
61 1−402.T+2.26e5T2 1 - 402.T + 2.26e5T^{2}
67 1+114.T+3.00e5T2 1 + 114.T + 3.00e5T^{2}
71 1+214.T+3.57e5T2 1 + 214.T + 3.57e5T^{2}
73 1−402.T+3.89e5T2 1 - 402.T + 3.89e5T^{2}
79 1+1.37e3T+4.93e5T2 1 + 1.37e3T + 4.93e5T^{2}
83 1+1.15e3T+5.71e5T2 1 + 1.15e3T + 5.71e5T^{2}
89 1+366.T+7.04e5T2 1 + 366.T + 7.04e5T^{2}
97 1+1.06e3T+9.12e5T2 1 + 1.06e3T + 9.12e5T^{2}
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   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

−10.86346312755941098979034917844, −9.663395962957927789884371756961, −8.724989337689837375123592810194, −8.259430965560902117938637901779, −6.83508950229635540164151879136, −5.77795834298411916829247256156, −4.26248153342305032211607885846, −3.87528630901471435766539165442, −1.66079032692313443558990512394, 0, 1.66079032692313443558990512394, 3.87528630901471435766539165442, 4.26248153342305032211607885846, 5.77795834298411916829247256156, 6.83508950229635540164151879136, 8.259430965560902117938637901779, 8.724989337689837375123592810194, 9.663395962957927789884371756961, 10.86346312755941098979034917844

Graph of the ZZ-function along the critical line