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Properties

Label 2-234-117.41-c1-0-3
Degree 22
Conductor 234234
Sign 0.550−0.834i0.550 - 0.834i
Analytic cond. 1.868491.86849
Root an. cond. 1.366931.36693
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 + 0.258i)2-s + (0.880 + 1.49i)3-s + (0.866 − 0.499i)4-s + (1.93 − 0.517i)5-s + (−1.23 − 1.21i)6-s + (1.87 + 1.87i)7-s + (−0.707 + 0.707i)8-s + (−1.44 + 2.62i)9-s + (−1.73 + 0.999i)10-s + (−1.12 − 4.21i)11-s + (1.50 + 0.851i)12-s + (1.69 − 3.18i)13-s + (−2.29 − 1.32i)14-s + (2.47 + 2.42i)15-s + (0.500 − 0.866i)16-s + (−1.21 + 2.09i)17-s + ⋯
L(s)  = 1  + (−0.683 + 0.183i)2-s + (0.508 + 0.861i)3-s + (0.433 − 0.249i)4-s + (0.863 − 0.231i)5-s + (−0.504 − 0.495i)6-s + (0.709 + 0.709i)7-s + (−0.249 + 0.249i)8-s + (−0.482 + 0.875i)9-s + (−0.547 + 0.316i)10-s + (−0.340 − 1.27i)11-s + (0.435 + 0.245i)12-s + (0.470 − 0.882i)13-s + (−0.614 − 0.354i)14-s + (0.638 + 0.625i)15-s + (0.125 − 0.216i)16-s + (−0.293 + 0.508i)17-s + ⋯

Functional equation

Λ(s)=(234s/2 ΓC(s) L(s)=((0.550−0.834i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(234s/2 ΓC(s+1/2) L(s)=((0.550−0.834i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 234 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.550 - 0.834i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 234234    =    2⋅32⋅132 \cdot 3^{2} \cdot 13
Sign: 0.550−0.834i0.550 - 0.834i
Analytic conductor: 1.868491.86849
Root analytic conductor: 1.366931.36693
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ234(41,⋅)\chi_{234} (41, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 234, ( :1/2), 0.550−0.834i)(2,\ 234,\ (\ :1/2),\ 0.550 - 0.834i)

Particular Values

L(1)L(1) ≈\approx 1.09605+0.590340i1.09605 + 0.590340i
L(12)L(\frac12) ≈\approx 1.09605+0.590340i1.09605 + 0.590340i
L(32)L(\frac{3}{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)
bad2 1+(0.965−0.258i)T 1 + (0.965 - 0.258i)T
3 1+(−0.880−1.49i)T 1 + (-0.880 - 1.49i)T
13 1+(−1.69+3.18i)T 1 + (-1.69 + 3.18i)T
good5 1+(−1.93+0.517i)T+(4.33−2.5i)T2 1 + (-1.93 + 0.517i)T + (4.33 - 2.5i)T^{2}
7 1+(−1.87−1.87i)T+7iT2 1 + (-1.87 - 1.87i)T + 7iT^{2}
11 1+(1.12+4.21i)T+(−9.52+5.5i)T2 1 + (1.12 + 4.21i)T + (-9.52 + 5.5i)T^{2}
17 1+(1.21−2.09i)T+(−8.5−14.7i)T2 1 + (1.21 - 2.09i)T + (-8.5 - 14.7i)T^{2}
19 1+(−1.64−6.14i)T+(−16.4+9.5i)T2 1 + (-1.64 - 6.14i)T + (-16.4 + 9.5i)T^{2}
23 1+1.22T+23T2 1 + 1.22T + 23T^{2}
29 1+(3.87+2.23i)T+(14.5+25.1i)T2 1 + (3.87 + 2.23i)T + (14.5 + 25.1i)T^{2}
31 1+(−2.18−8.14i)T+(−26.8+15.5i)T2 1 + (-2.18 - 8.14i)T + (-26.8 + 15.5i)T^{2}
37 1+(−2.49+9.31i)T+(−32.0−18.5i)T2 1 + (-2.49 + 9.31i)T + (-32.0 - 18.5i)T^{2}
41 1+(4.85+4.85i)T+41iT2 1 + (4.85 + 4.85i)T + 41iT^{2}
43 1+10.2iT−43T2 1 + 10.2iT - 43T^{2}
47 1+(−5.22−1.40i)T+(40.7+23.5i)T2 1 + (-5.22 - 1.40i)T + (40.7 + 23.5i)T^{2}
53 1+0.142iT−53T2 1 + 0.142iT - 53T^{2}
59 1+(3.88+1.03i)T+(51.0+29.5i)T2 1 + (3.88 + 1.03i)T + (51.0 + 29.5i)T^{2}
61 1+10.0T+61T2 1 + 10.0T + 61T^{2}
67 1+(−1.78+1.78i)T−67iT2 1 + (-1.78 + 1.78i)T - 67iT^{2}
71 1+(−7.11+1.90i)T+(61.4−35.5i)T2 1 + (-7.11 + 1.90i)T + (61.4 - 35.5i)T^{2}
73 1+(4.15+4.15i)T+73iT2 1 + (4.15 + 4.15i)T + 73iT^{2}
79 1+(0.161+0.280i)T+(−39.5+68.4i)T2 1 + (0.161 + 0.280i)T + (-39.5 + 68.4i)T^{2}
83 1+(1.01−3.79i)T+(−71.8−41.5i)T2 1 + (1.01 - 3.79i)T + (-71.8 - 41.5i)T^{2}
89 1+(−14.2−3.83i)T+(77.0+44.5i)T2 1 + (-14.2 - 3.83i)T + (77.0 + 44.5i)T^{2}
97 1+(6.20−6.20i)T−97iT2 1 + (6.20 - 6.20i)T - 97iT^{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

−12.17144078119423542449976571220, −10.87047670898289381622928179796, −10.40798650185799470883882653388, −9.257405934875122971736873049578, −8.535018565610980831006751243938, −7.88747123987019399893579451016, −5.79568561360711879143764641523, −5.47032954551496028519042850845, −3.50874713788117291724269791025, −1.98957473353963113512414796865, 1.52649536209657578202572529442, 2.59284645692781251545966189280, 4.54489396010783974622225322195, 6.34185137391732751163705904837, 7.18506627708218066799612283670, 7.968955856551595116311361922642, 9.240522984711386604644401208749, 9.826794962996286530995221311902, 11.12920115055266462306247378399, 11.83934454387743072145095413194

Graph of the ZZ-function along the critical line