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Properties

Label 2-231-1.1-c3-0-2
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  − 4.79·2-s − 3·3-s + 14.9·4-s − 6.25·5-s + 14.3·6-s + 7·7-s − 33.4·8-s + 9·9-s + 29.9·10-s + 11·11-s − 44.9·12-s − 35.7·13-s − 33.5·14-s + 18.7·15-s + 40.4·16-s − 133.·17-s − 43.1·18-s + 161.·19-s − 93.6·20-s − 21·21-s − 52.7·22-s − 66.2·23-s + 100.·24-s − 85.8·25-s + 171.·26-s − 27·27-s + 104.·28-s + ⋯
L(s)  = 1  − 1.69·2-s − 0.577·3-s + 1.87·4-s − 0.559·5-s + 0.978·6-s + 0.377·7-s − 1.47·8-s + 0.333·9-s + 0.947·10-s + 0.301·11-s − 1.08·12-s − 0.763·13-s − 0.640·14-s + 0.322·15-s + 0.632·16-s − 1.89·17-s − 0.564·18-s + 1.95·19-s − 1.04·20-s − 0.218·21-s − 0.510·22-s − 0.600·23-s + 0.853·24-s − 0.687·25-s + 1.29·26-s − 0.192·27-s + 0.707·28-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 0.44726319160.4472631916
L(12)L(\frac12) ≈\approx 0.44726319160.4472631916
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+4.79T+8T2 1 + 4.79T + 8T^{2}
5 1+6.25T+125T2 1 + 6.25T + 125T^{2}
13 1+35.7T+2.19e3T2 1 + 35.7T + 2.19e3T^{2}
17 1+133.T+4.91e3T2 1 + 133.T + 4.91e3T^{2}
19 1−161.T+6.85e3T2 1 - 161.T + 6.85e3T^{2}
23 1+66.2T+1.21e4T2 1 + 66.2T + 1.21e4T^{2}
29 1+208.T+2.43e4T2 1 + 208.T + 2.43e4T^{2}
31 1+39.3T+2.97e4T2 1 + 39.3T + 2.97e4T^{2}
37 1−197.T+5.06e4T2 1 - 197.T + 5.06e4T^{2}
41 1−434.T+6.89e4T2 1 - 434.T + 6.89e4T^{2}
43 1−375.T+7.95e4T2 1 - 375.T + 7.95e4T^{2}
47 1−503.T+1.03e5T2 1 - 503.T + 1.03e5T^{2}
53 1−44.8T+1.48e5T2 1 - 44.8T + 1.48e5T^{2}
59 1−582.T+2.05e5T2 1 - 582.T + 2.05e5T^{2}
61 1−73.2T+2.26e5T2 1 - 73.2T + 2.26e5T^{2}
67 1+928.T+3.00e5T2 1 + 928.T + 3.00e5T^{2}
71 1+755.T+3.57e5T2 1 + 755.T + 3.57e5T^{2}
73 1−277.T+3.89e5T2 1 - 277.T + 3.89e5T^{2}
79 1−651.T+4.93e5T2 1 - 651.T + 4.93e5T^{2}
83 1−282.T+5.71e5T2 1 - 282.T + 5.71e5T^{2}
89 1−1.04e3T+7.04e5T2 1 - 1.04e3T + 7.04e5T^{2}
97 1−1.11e3T+9.12e5T2 1 - 1.11e3T + 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

−11.42021745880362183088851150471, −10.81204534302661965064055944734, −9.613754558183303599836511308971, −9.020522549620414851049524799866, −7.63251803125824198541241949753, −7.29948424442187797992717648552, −5.87878725079517622421917049697, −4.29054404812829399476248449579, −2.20427854497176308652515261058, −0.63991171926699498756646470719, 0.63991171926699498756646470719, 2.20427854497176308652515261058, 4.29054404812829399476248449579, 5.87878725079517622421917049697, 7.29948424442187797992717648552, 7.63251803125824198541241949753, 9.020522549620414851049524799866, 9.613754558183303599836511308971, 10.81204534302661965064055944734, 11.42021745880362183088851150471

Graph of the ZZ-function along the critical line