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Properties

Label 2-231-1.1-c3-0-4
Degree 22
Conductor 231231
Sign 11
Analytic cond. 13.629413.6294
Root an. cond. 3.691803.69180
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.300·2-s + 3·3-s − 7.90·4-s − 20.3·5-s − 0.900·6-s − 7·7-s + 4.77·8-s + 9·9-s + 6.12·10-s + 11·11-s − 23.7·12-s + 14.2·13-s + 2.10·14-s − 61.1·15-s + 61.8·16-s + 137.·17-s − 2.70·18-s − 44.1·19-s + 161.·20-s − 21·21-s − 3.30·22-s + 71.7·23-s + 14.3·24-s + 290.·25-s − 4.26·26-s + 27·27-s + 55.3·28-s + ⋯
L(s)  = 1  − 0.106·2-s + 0.577·3-s − 0.988·4-s − 1.82·5-s − 0.0612·6-s − 0.377·7-s + 0.211·8-s + 0.333·9-s + 0.193·10-s + 0.301·11-s − 0.570·12-s + 0.303·13-s + 0.0401·14-s − 1.05·15-s + 0.966·16-s + 1.96·17-s − 0.0353·18-s − 0.533·19-s + 1.80·20-s − 0.218·21-s − 0.0319·22-s + 0.650·23-s + 0.121·24-s + 2.32·25-s − 0.0321·26-s + 0.192·27-s + 0.373·28-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=( Λ(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+3/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 13.629413.6294
Root analytic conductor: 3.691803.69180
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 231, ( :3/2), 1)(2,\ 231,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) ≈\approx 1.0368167661.036816766
L(12)L(\frac12) ≈\approx 1.0368167661.036816766
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 1−3T 1 - 3T
7 1+7T 1 + 7T
11 1−11T 1 - 11T
good2 1+0.300T+8T2 1 + 0.300T + 8T^{2}
5 1+20.3T+125T2 1 + 20.3T + 125T^{2}
13 1−14.2T+2.19e3T2 1 - 14.2T + 2.19e3T^{2}
17 1−137.T+4.91e3T2 1 - 137.T + 4.91e3T^{2}
19 1+44.1T+6.85e3T2 1 + 44.1T + 6.85e3T^{2}
23 1−71.7T+1.21e4T2 1 - 71.7T + 1.21e4T^{2}
29 1+59.5T+2.43e4T2 1 + 59.5T + 2.43e4T^{2}
31 1+207.T+2.97e4T2 1 + 207.T + 2.97e4T^{2}
37 1−126.T+5.06e4T2 1 - 126.T + 5.06e4T^{2}
41 1+283.T+6.89e4T2 1 + 283.T + 6.89e4T^{2}
43 1+179.T+7.95e4T2 1 + 179.T + 7.95e4T^{2}
47 1−94.8T+1.03e5T2 1 - 94.8T + 1.03e5T^{2}
53 1−727.T+1.48e5T2 1 - 727.T + 1.48e5T^{2}
59 1−677.T+2.05e5T2 1 - 677.T + 2.05e5T^{2}
61 1−210.T+2.26e5T2 1 - 210.T + 2.26e5T^{2}
67 1+428.T+3.00e5T2 1 + 428.T + 3.00e5T^{2}
71 1−419.T+3.57e5T2 1 - 419.T + 3.57e5T^{2}
73 1−921.T+3.89e5T2 1 - 921.T + 3.89e5T^{2}
79 1−895.T+4.93e5T2 1 - 895.T + 4.93e5T^{2}
83 1−1.18e3T+5.71e5T2 1 - 1.18e3T + 5.71e5T^{2}
89 1−8.28T+7.04e5T2 1 - 8.28T + 7.04e5T^{2}
97 1+828.T+9.12e5T2 1 + 828.T + 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

−11.95558528586642572122588559467, −10.74123681164674524083411686850, −9.636148570415690384731233758894, −8.626543140402445642647155194021, −7.980475411515572580928149548201, −7.09221741471787987251017554944, −5.24746677805671612354394371611, −3.90085668211618254797063622590, −3.43984794539157916794469039252, −0.76048143786217510814193532829, 0.76048143786217510814193532829, 3.43984794539157916794469039252, 3.90085668211618254797063622590, 5.24746677805671612354394371611, 7.09221741471787987251017554944, 7.980475411515572580928149548201, 8.626543140402445642647155194021, 9.636148570415690384731233758894, 10.74123681164674524083411686850, 11.95558528586642572122588559467

Graph of the ZZ-function along the critical line