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Properties

Label 2-231-11.3-c3-0-21
Degree 22
Conductor 231231
Sign 0.711+0.702i0.711 + 0.702i
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  + (−0.992 + 0.721i)2-s + (−0.927 − 2.85i)3-s + (−2.00 + 6.17i)4-s + (2.71 + 1.97i)5-s + (2.97 + 2.16i)6-s + (2.16 − 6.65i)7-s + (−5.49 − 16.9i)8-s + (−7.28 + 5.29i)9-s − 4.12·10-s + (−36.1 − 4.60i)11-s + 19.4·12-s + (35.9 − 26.1i)13-s + (2.65 + 8.16i)14-s + (3.11 − 9.58i)15-s + (−24.3 − 17.7i)16-s + (26.8 + 19.5i)17-s + ⋯
L(s)  = 1  + (−0.350 + 0.254i)2-s + (−0.178 − 0.549i)3-s + (−0.250 + 0.772i)4-s + (0.243 + 0.176i)5-s + (0.202 + 0.147i)6-s + (0.116 − 0.359i)7-s + (−0.242 − 0.747i)8-s + (−0.269 + 0.195i)9-s − 0.130·10-s + (−0.991 − 0.126i)11-s + 0.468·12-s + (0.766 − 0.556i)13-s + (0.0506 + 0.155i)14-s + (0.0536 − 0.165i)15-s + (−0.380 − 0.276i)16-s + (0.383 + 0.278i)17-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=((0.711+0.702i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+3/2) L(s)=((0.711+0.702i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.711 + 0.702i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 0.711+0.702i0.711 + 0.702i
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), 0.711+0.702i)(2,\ 231,\ (\ :3/2),\ 0.711 + 0.702i)

Particular Values

L(2)L(2) ≈\approx 1.03234−0.423584i1.03234 - 0.423584i
L(12)L(\frac12) ≈\approx 1.03234−0.423584i1.03234 - 0.423584i
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+(36.1+4.60i)T 1 + (36.1 + 4.60i)T
good2 1+(0.992−0.721i)T+(2.47−7.60i)T2 1 + (0.992 - 0.721i)T + (2.47 - 7.60i)T^{2}
5 1+(−2.71−1.97i)T+(38.6+118.i)T2 1 + (-2.71 - 1.97i)T + (38.6 + 118. i)T^{2}
13 1+(−35.9+26.1i)T+(678.−2.08e3i)T2 1 + (-35.9 + 26.1i)T + (678. - 2.08e3i)T^{2}
17 1+(−26.8−19.5i)T+(1.51e3+4.67e3i)T2 1 + (-26.8 - 19.5i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(−21.1−65.1i)T+(−5.54e3+4.03e3i)T2 1 + (-21.1 - 65.1i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−65.7T+1.21e4T2 1 - 65.7T + 1.21e4T^{2}
29 1+(−71.7+220.i)T+(−1.97e4−1.43e4i)T2 1 + (-71.7 + 220. i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(−222.+161.i)T+(9.20e3−2.83e4i)T2 1 + (-222. + 161. i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−28.4+87.6i)T+(−4.09e4−2.97e4i)T2 1 + (-28.4 + 87.6i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(72.0+221.i)T+(−5.57e4+4.05e4i)T2 1 + (72.0 + 221. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−312.T+7.95e4T2 1 - 312.T + 7.95e4T^{2}
47 1+(107.+329.i)T+(−8.39e4+6.10e4i)T2 1 + (107. + 329. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(589.−428.i)T+(4.60e4−1.41e5i)T2 1 + (589. - 428. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−215.+664.i)T+(−1.66e5−1.20e5i)T2 1 + (-215. + 664. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−509.−370.i)T+(7.01e4+2.15e5i)T2 1 + (-509. - 370. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1−1.09e3T+3.00e5T2 1 - 1.09e3T + 3.00e5T^{2}
71 1+(125.+91.0i)T+(1.10e5+3.40e5i)T2 1 + (125. + 91.0i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(132.−408.i)T+(−3.14e5−2.28e5i)T2 1 + (132. - 408. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(216.−157.i)T+(1.52e5−4.68e5i)T2 1 + (216. - 157. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(412.+299.i)T+(1.76e5+5.43e5i)T2 1 + (412. + 299. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1−336.T+7.04e5T2 1 - 336.T + 7.04e5T^{2}
97 1+(844.−613.i)T+(2.82e5−8.68e5i)T2 1 + (844. - 613. 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

−11.73833186568670048787825574289, −10.59666332297843147032453725847, −9.698763840605597378256135957508, −8.084720218668722568735417639421, −8.049155543698125403573515992906, −6.68574923210715411631755735674, −5.62985483899360334683928794269, −4.02036405449223873756691852379, −2.64800920015042330910806541821, −0.61478326401253917265736106591, 1.21893030690451184679945851773, 2.89551948313642977045980857811, 4.78484584843483998975453201971, 5.40519749643122449454399623835, 6.64248839427536515585785563829, 8.289786305467978316028018739532, 9.157000944692769059148598593778, 9.900863209099034932011388646940, 10.86771511601175594970209825907, 11.47032874445879945562619127569

Graph of the ZZ-function along the critical line