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Properties

Label 2-231-11.3-c3-0-26
Degree 22
Conductor 231231
Sign 0.990−0.140i0.990 - 0.140i
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  + (−2.72 + 1.97i)2-s + (0.927 + 2.85i)3-s + (1.03 − 3.18i)4-s + (6.94 + 5.04i)5-s + (−8.17 − 5.93i)6-s + (2.16 − 6.65i)7-s + (−4.84 − 14.9i)8-s + (−7.28 + 5.29i)9-s − 28.9·10-s + (25.6 − 25.9i)11-s + 10.0·12-s + (65.0 − 47.2i)13-s + (7.28 + 22.4i)14-s + (−7.95 + 24.4i)15-s + (64.3 + 46.7i)16-s + (−98.4 − 71.5i)17-s + ⋯
L(s)  = 1  + (−0.963 + 0.699i)2-s + (0.178 + 0.549i)3-s + (0.129 − 0.397i)4-s + (0.621 + 0.451i)5-s + (−0.556 − 0.404i)6-s + (0.116 − 0.359i)7-s + (−0.214 − 0.658i)8-s + (−0.269 + 0.195i)9-s − 0.914·10-s + (0.701 − 0.712i)11-s + 0.241·12-s + (1.38 − 1.00i)13-s + (0.139 + 0.428i)14-s + (−0.136 + 0.421i)15-s + (1.00 + 0.730i)16-s + (−1.40 − 1.02i)17-s + ⋯

Functional equation

Λ(s)=(231s/2 ΓC(s) L(s)=((0.990−0.140i) Λ‾(4−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(4-s) \end{aligned}
Λ(s)=(231s/2 ΓC(s+3/2) L(s)=((0.990−0.140i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 231 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.990 - 0.140i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 231231    =    3⋅7⋅113 \cdot 7 \cdot 11
Sign: 0.990−0.140i0.990 - 0.140i
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), 0.990−0.140i)(2,\ 231,\ (\ :3/2),\ 0.990 - 0.140i)

Particular Values

L(2)L(2) ≈\approx 1.10286+0.0777048i1.10286 + 0.0777048i
L(12)L(\frac12) ≈\approx 1.10286+0.0777048i1.10286 + 0.0777048i
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+(−25.6+25.9i)T 1 + (-25.6 + 25.9i)T
good2 1+(2.72−1.97i)T+(2.47−7.60i)T2 1 + (2.72 - 1.97i)T + (2.47 - 7.60i)T^{2}
5 1+(−6.94−5.04i)T+(38.6+118.i)T2 1 + (-6.94 - 5.04i)T + (38.6 + 118. i)T^{2}
13 1+(−65.0+47.2i)T+(678.−2.08e3i)T2 1 + (-65.0 + 47.2i)T + (678. - 2.08e3i)T^{2}
17 1+(98.4+71.5i)T+(1.51e3+4.67e3i)T2 1 + (98.4 + 71.5i)T + (1.51e3 + 4.67e3i)T^{2}
19 1+(41.5+127.i)T+(−5.54e3+4.03e3i)T2 1 + (41.5 + 127. i)T + (-5.54e3 + 4.03e3i)T^{2}
23 1−9.39T+1.21e4T2 1 - 9.39T + 1.21e4T^{2}
29 1+(−12.2+37.6i)T+(−1.97e4−1.43e4i)T2 1 + (-12.2 + 37.6i)T + (-1.97e4 - 1.43e4i)T^{2}
31 1+(21.8−15.8i)T+(9.20e3−2.83e4i)T2 1 + (21.8 - 15.8i)T + (9.20e3 - 2.83e4i)T^{2}
37 1+(−107.+329.i)T+(−4.09e4−2.97e4i)T2 1 + (-107. + 329. i)T + (-4.09e4 - 2.97e4i)T^{2}
41 1+(−134.−413.i)T+(−5.57e4+4.05e4i)T2 1 + (-134. - 413. i)T + (-5.57e4 + 4.05e4i)T^{2}
43 1−51.0T+7.95e4T2 1 - 51.0T + 7.95e4T^{2}
47 1+(−132.−406.i)T+(−8.39e4+6.10e4i)T2 1 + (-132. - 406. i)T + (-8.39e4 + 6.10e4i)T^{2}
53 1+(430.−312.i)T+(4.60e4−1.41e5i)T2 1 + (430. - 312. i)T + (4.60e4 - 1.41e5i)T^{2}
59 1+(−154.+475.i)T+(−1.66e5−1.20e5i)T2 1 + (-154. + 475. i)T + (-1.66e5 - 1.20e5i)T^{2}
61 1+(−226.−164.i)T+(7.01e4+2.15e5i)T2 1 + (-226. - 164. i)T + (7.01e4 + 2.15e5i)T^{2}
67 1+145.T+3.00e5T2 1 + 145.T + 3.00e5T^{2}
71 1+(45.4+33.0i)T+(1.10e5+3.40e5i)T2 1 + (45.4 + 33.0i)T + (1.10e5 + 3.40e5i)T^{2}
73 1+(−78.4+241.i)T+(−3.14e5−2.28e5i)T2 1 + (-78.4 + 241. i)T + (-3.14e5 - 2.28e5i)T^{2}
79 1+(−260.+188.i)T+(1.52e5−4.68e5i)T2 1 + (-260. + 188. i)T + (1.52e5 - 4.68e5i)T^{2}
83 1+(347.+252.i)T+(1.76e5+5.43e5i)T2 1 + (347. + 252. i)T + (1.76e5 + 5.43e5i)T^{2}
89 1+636.T+7.04e5T2 1 + 636.T + 7.04e5T^{2}
97 1+(−1.34e3+979.i)T+(2.82e5−8.68e5i)T2 1 + (-1.34e3 + 979. 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

−11.16446285175321323506115991229, −10.76721163159528796914081810464, −9.449014829414120265122812310270, −8.952895512990913764788739268793, −7.985904185577284965067053362327, −6.72770447291987879105032576966, −6.02159460123983235997693638109, −4.29094329920643348827903974096, −2.90677091615609957069865803946, −0.64999170425687709831318753249, 1.50237329657810055092943555537, 1.95544186373615466844308772915, 4.01658835956333103624552222434, 5.75007416284251125633145318207, 6.66628928337810556522661167562, 8.339383314166130830350161130968, 8.820619542530787980155424953463, 9.638408109391464886717530447851, 10.72695764256582608586271770235, 11.60129463469179839497127075376

Graph of the ZZ-function along the critical line