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Properties

Label 2-29e2-29.20-c1-0-10
Degree 22
Conductor 841841
Sign 0.973−0.230i0.973 - 0.230i
Analytic cond. 6.715416.71541
Root an. cond. 2.591412.59141
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  + (−0.556 + 0.268i)2-s + (−0.385 − 0.483i)3-s + (−1.00 + 1.26i)4-s + (−3.47 + 1.67i)5-s + (0.344 + 0.165i)6-s + (−1.39 − 1.74i)7-s + (0.497 − 2.18i)8-s + (0.582 − 2.55i)9-s + (1.48 − 1.86i)10-s + (0.307 + 1.34i)11-s + 12-s + (0.0525 + 0.230i)13-s + (1.24 + 0.599i)14-s + (2.14 + 1.03i)15-s + (−0.412 − 1.80i)16-s − 4.38·17-s + ⋯
L(s)  = 1  + (−0.393 + 0.189i)2-s + (−0.222 − 0.278i)3-s + (−0.504 + 0.632i)4-s + (−1.55 + 0.747i)5-s + (0.140 + 0.0676i)6-s + (−0.526 − 0.660i)7-s + (0.175 − 0.770i)8-s + (0.194 − 0.850i)9-s + (0.469 − 0.588i)10-s + (0.0927 + 0.406i)11-s + 0.288·12-s + (0.0145 + 0.0638i)13-s + (0.332 + 0.160i)14-s + (0.554 + 0.266i)15-s + (−0.103 − 0.451i)16-s − 1.06·17-s + ⋯

Functional equation

Λ(s)=(841s/2 ΓC(s) L(s)=((0.973−0.230i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(841s/2 ΓC(s+1/2) L(s)=((0.973−0.230i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 841 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 841841    =    29229^{2}
Sign: 0.973−0.230i0.973 - 0.230i
Analytic conductor: 6.715416.71541
Root analytic conductor: 2.591412.59141
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ841(571,⋅)\chi_{841} (571, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 841, ( :1/2), 0.973−0.230i)(2,\ 841,\ (\ :1/2),\ 0.973 - 0.230i)

Particular Values

L(1)L(1) ≈\approx 0.492690+0.0575577i0.492690 + 0.0575577i
L(12)L(\frac12) ≈\approx 0.492690+0.0575577i0.492690 + 0.0575577i
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)
bad29 1 1
good2 1+(0.556−0.268i)T+(1.24−1.56i)T2 1 + (0.556 - 0.268i)T + (1.24 - 1.56i)T^{2}
3 1+(0.385+0.483i)T+(−0.667+2.92i)T2 1 + (0.385 + 0.483i)T + (-0.667 + 2.92i)T^{2}
5 1+(3.47−1.67i)T+(3.11−3.90i)T2 1 + (3.47 - 1.67i)T + (3.11 - 3.90i)T^{2}
7 1+(1.39+1.74i)T+(−1.55+6.82i)T2 1 + (1.39 + 1.74i)T + (-1.55 + 6.82i)T^{2}
11 1+(−0.307−1.34i)T+(−9.91+4.77i)T2 1 + (-0.307 - 1.34i)T + (-9.91 + 4.77i)T^{2}
13 1+(−0.0525−0.230i)T+(−11.7+5.64i)T2 1 + (-0.0525 - 0.230i)T + (-11.7 + 5.64i)T^{2}
17 1+4.38T+17T2 1 + 4.38T + 17T^{2}
19 1+(3.02−3.79i)T+(−4.22−18.5i)T2 1 + (3.02 - 3.79i)T + (-4.22 - 18.5i)T^{2}
23 1+(−1.11−0.536i)T+(14.3+17.9i)T2 1 + (-1.11 - 0.536i)T + (14.3 + 17.9i)T^{2}
31 1+(−9.09+4.37i)T+(19.3−24.2i)T2 1 + (-9.09 + 4.37i)T + (19.3 - 24.2i)T^{2}
37 1+(1.04−4.59i)T+(−33.3−16.0i)T2 1 + (1.04 - 4.59i)T + (-33.3 - 16.0i)T^{2}
41 1−3.85T+41T2 1 - 3.85T + 41T^{2}
43 1+(−6.51−3.13i)T+(26.8+33.6i)T2 1 + (-6.51 - 3.13i)T + (26.8 + 33.6i)T^{2}
47 1+(−1.55−6.82i)T+(−42.3+20.3i)T2 1 + (-1.55 - 6.82i)T + (-42.3 + 20.3i)T^{2}
53 1+(−1.80+0.867i)T+(33.0−41.4i)T2 1 + (-1.80 + 0.867i)T + (33.0 - 41.4i)T^{2}
59 1−6.09T+59T2 1 - 6.09T + 59T^{2}
61 1+(0.385+0.483i)T+(−13.5+59.4i)T2 1 + (0.385 + 0.483i)T + (-13.5 + 59.4i)T^{2}
67 1+(−0.339+1.48i)T+(−60.3−29.0i)T2 1 + (-0.339 + 1.48i)T + (-60.3 - 29.0i)T^{2}
71 1+(2.33+10.2i)T+(−63.9+30.8i)T2 1 + (2.33 + 10.2i)T + (-63.9 + 30.8i)T^{2}
73 1+(−12.3−5.94i)T+(45.5+57.0i)T2 1 + (-12.3 - 5.94i)T + (45.5 + 57.0i)T^{2}
79 1+(−1.35+5.93i)T+(−71.1−34.2i)T2 1 + (-1.35 + 5.93i)T + (-71.1 - 34.2i)T^{2}
83 1+(6.20−7.77i)T+(−18.4−80.9i)T2 1 + (6.20 - 7.77i)T + (-18.4 - 80.9i)T^{2}
89 1+(4.24−2.04i)T+(55.4−69.5i)T2 1 + (4.24 - 2.04i)T + (55.4 - 69.5i)T^{2}
97 1+(−2.22+2.78i)T+(−21.5−94.5i)T2 1 + (-2.22 + 2.78i)T + (-21.5 - 94.5i)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

−10.17333468567448912095392189655, −9.341239925464861039400647367827, −8.308593250661852781774420457237, −7.68056195013240521035390817190, −6.87732693963300807417524174889, −6.43572562836321539502692952725, −4.28712737180342224394427025342, −4.03784564238360633543929468019, −2.98507744484102649545987952670, −0.56226121772665524297482685135, 0.63118108153410745310510595568, 2.46486924639989070843017133410, 4.08700952240942390392367118532, 4.69684914474254772634402347461, 5.53491290433759087902666325059, 6.76561445005449166260024544599, 7.936161182267056858643723640835, 8.735486677518784825368171223503, 9.028874431407587236598256803753, 10.26083555455323772640853894130

Graph of the ZZ-function along the critical line