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Properties

Label 2-231-11.3-c3-0-12
Degree 22
Conductor 231231
Sign −0.612−0.790i-0.612 - 0.790i
Analytic cond. 13.629413.6294
Root an. cond. 3.691803.69180
Motivic weight 33
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.446 − 0.324i)2-s + (0.927 + 2.85i)3-s + (−2.37 + 7.31i)4-s + (9.46 + 6.87i)5-s + (1.33 + 0.972i)6-s + (−2.16 + 6.65i)7-s + (2.67 + 8.23i)8-s + (−7.28 + 5.29i)9-s + 6.45·10-s + (35.2 + 9.37i)11-s − 23.0·12-s + (−7.29 + 5.30i)13-s + (1.19 + 3.67i)14-s + (−10.8 + 33.3i)15-s + (−45.9 − 33.3i)16-s + (17.8 + 12.9i)17-s + ⋯
L(s)  = 1  + (0.157 − 0.114i)2-s + (0.178 + 0.549i)3-s + (−0.297 + 0.914i)4-s + (0.846 + 0.615i)5-s + (0.0911 + 0.0661i)6-s + (−0.116 + 0.359i)7-s + (0.118 + 0.363i)8-s + (−0.269 + 0.195i)9-s + 0.204·10-s + (0.966 + 0.256i)11-s − 0.555·12-s + (−0.155 + 0.113i)13-s + (0.0227 + 0.0701i)14-s + (−0.186 + 0.574i)15-s + (−0.717 − 0.521i)16-s + (0.254 + 0.184i)17-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=((−0.612−0.790i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+3/2) L(s)=((−0.612−0.790i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.612 - 0.790i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: −0.612−0.790i-0.612 - 0.790i
Analytic conductor: 13.629413.6294
Root analytic conductor: 3.691803.69180
Motivic weight: 33
Rational: no
Arithmetic: yes
Character: χ231(190,⋅)\chi_{231} (190, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 231, ( :3/2), −0.612−0.790i)(2,\ 231,\ (\ :3/2),\ -0.612 - 0.790i)

Particular Values

L(2)L(2) ≈\approx 0.890719+1.81622i0.890719 + 1.81622i
L(12)L(\frac12) ≈\approx 0.890719+1.81622i0.890719 + 1.81622i
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+(−0.927−2.85i)T 1 + (-0.927 - 2.85i)T
7 1+(2.16−6.65i)T 1 + (2.16 - 6.65i)T
11 1+(−35.2−9.37i)T 1 + (-35.2 - 9.37i)T
good2 1+(−0.446+0.324i)T+(2.47−7.60i)T2 1 + (-0.446 + 0.324i)T + (2.47 - 7.60i)T^{2}
5 1+(−9.46−6.87i)T+(38.6+118.i)T2 1 + (-9.46 - 6.87i)T + (38.6 + 118. i)T^{2}
13 1+(7.29−5.30i)T+(678.−2.08e3i)T2 1 + (7.29 - 5.30i)T + (678. - 2.08e3i)T^{2}
17 1+(−17.8−12.9i)T+(1.51e3+4.67e3i)T2 1 + (-17.8 - 12.9i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(5.29+16.3i)T+(−5.54e3+4.03e3i)T2 1 + (5.29 + 16.3i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1+69.0T+1.21e4T2 1 + 69.0T + 1.21e4T^{2}
29 1+(−6.39+19.6i)T+(−1.97e4−1.43e4i)T2 1 + (-6.39 + 19.6i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(89.7−65.2i)T+(9.20e3−2.83e4i)T2 1 + (89.7 - 65.2i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(32.0−98.6i)T+(−4.09e4−2.97e4i)T2 1 + (32.0 - 98.6i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−31.8−98.0i)T+(−5.57e4+4.05e4i)T2 1 + (-31.8 - 98.0i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−170.T+7.95e4T2 1 - 170.T + 7.95e4T^{2}
47 1+(−93.0−286.i)T+(−8.39e4+6.10e4i)T2 1 + (-93.0 - 286. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(57.0−41.4i)T+(4.60e4−1.41e5i)T2 1 + (57.0 - 41.4i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−194.+599.i)T+(−1.66e5−1.20e5i)T2 1 + (-194. + 599. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−541.−393.i)T+(7.01e4+2.15e5i)T2 1 + (-541. - 393. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−653.T+3.00e5T2 1 - 653.T + 3.00e5T^{2}
71 1+(525.+381.i)T+(1.10e5+3.40e5i)T2 1 + (525. + 381. i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−33.4+102.i)T+(−3.14e5−2.28e5i)T2 1 + (-33.4 + 102. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(409.−297.i)T+(1.52e5−4.68e5i)T2 1 + (409. - 297. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(55.2+40.1i)T+(1.76e5+5.43e5i)T2 1 + (55.2 + 40.1i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+337.T+7.04e5T2 1 + 337.T + 7.04e5T^{2}
97 1+(−551.+400.i)T+(2.82e5−8.68e5i)T2 1 + (-551. + 400. i)T + (2.82e5 - 8.68e5i)T^{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

−12.10295650452472430536258151754, −11.19480135110728817496700087254, −9.995297332327203315471834013883, −9.281362397389747340238875522146, −8.322884030807921405979746646266, −7.03668826037845681620216184824, −5.91462201383556693638299027211, −4.52239113021580842668951216785, −3.39941096718401197331260586270, −2.21530231553164432425640994422, 0.793759441732570591006067861387, 1.90983642047373156784121195562, 3.96131934551655119537486319606, 5.36652464527079758852245637880, 6.12332927921054366465847740405, 7.19053959848912613502865416008, 8.667804184526058781595787394077, 9.448028539862748958929752858936, 10.23385503319091800840478831964, 11.45839952140012230898999391910

Graph of the ZZ-function along the critical line