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Properties

Label 2-832-104.101-c1-0-14
Degree 22
Conductor 832832
Sign 0.246+0.969i0.246 + 0.969i
Analytic cond. 6.643556.64355
Root an. cond. 2.577502.57750
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  + (−0.866 + 0.5i)3-s + (−3.12 − 1.80i)7-s + (−1 + 1.73i)9-s + (0.866 + 1.5i)11-s + 3.60·13-s + (1.5 − 2.59i)17-s + (0.866 − 1.5i)19-s + 3.60·21-s + (−3.12 − 5.40i)23-s − 5·25-s − 5i·27-s + (5.40 − 3.12i)29-s − 7.21i·31-s + (−1.5 − 0.866i)33-s + (−1.80 − 3.12i)37-s + ⋯
L(s)  = 1  + (−0.499 + 0.288i)3-s + (−1.18 − 0.681i)7-s + (−0.333 + 0.577i)9-s + (0.261 + 0.452i)11-s + 1.00·13-s + (0.363 − 0.630i)17-s + (0.198 − 0.344i)19-s + 0.786·21-s + (−0.651 − 1.12i)23-s − 25-s − 0.962i·27-s + (1.00 − 0.579i)29-s − 1.29i·31-s + (−0.261 − 0.150i)33-s + (−0.296 − 0.513i)37-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.246+0.969i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.246+0.969i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.246 + 0.969i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.246+0.969i0.246 + 0.969i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(673,⋅)\chi_{832} (673, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.246+0.969i)(2,\ 832,\ (\ :1/2),\ 0.246 + 0.969i)

Particular Values

L(1)L(1) ≈\approx 0.647408−0.503401i0.647408 - 0.503401i
L(12)L(\frac12) ≈\approx 0.647408−0.503401i0.647408 - 0.503401i
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 1
13 1−3.60T 1 - 3.60T
good3 1+(0.866−0.5i)T+(1.5−2.59i)T2 1 + (0.866 - 0.5i)T + (1.5 - 2.59i)T^{2}
5 1+5T2 1 + 5T^{2}
7 1+(3.12+1.80i)T+(3.5+6.06i)T2 1 + (3.12 + 1.80i)T + (3.5 + 6.06i)T^{2}
11 1+(−0.866−1.5i)T+(−5.5+9.52i)T2 1 + (-0.866 - 1.5i)T + (-5.5 + 9.52i)T^{2}
17 1+(−1.5+2.59i)T+(−8.5−14.7i)T2 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.866+1.5i)T+(−9.5−16.4i)T2 1 + (-0.866 + 1.5i)T + (-9.5 - 16.4i)T^{2}
23 1+(3.12+5.40i)T+(−11.5+19.9i)T2 1 + (3.12 + 5.40i)T + (-11.5 + 19.9i)T^{2}
29 1+(−5.40+3.12i)T+(14.5−25.1i)T2 1 + (-5.40 + 3.12i)T + (14.5 - 25.1i)T^{2}
31 1+7.21iT−31T2 1 + 7.21iT - 31T^{2}
37 1+(1.80+3.12i)T+(−18.5+32.0i)T2 1 + (1.80 + 3.12i)T + (-18.5 + 32.0i)T^{2}
41 1+(−4.5+2.59i)T+(20.5−35.5i)T2 1 + (-4.5 + 2.59i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.866−0.5i)T+(21.5+37.2i)T2 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2}
47 1−47T2 1 - 47T^{2}
53 1+12.4iT−53T2 1 + 12.4iT - 53T^{2}
59 1+(6.06−10.5i)T+(−29.5−51.0i)T2 1 + (6.06 - 10.5i)T + (-29.5 - 51.0i)T^{2}
61 1+(−5.40−3.12i)T+(30.5+52.8i)T2 1 + (-5.40 - 3.12i)T + (30.5 + 52.8i)T^{2}
67 1+(0.866+1.5i)T+(−33.5+58.0i)T2 1 + (0.866 + 1.5i)T + (-33.5 + 58.0i)T^{2}
71 1+(9.36+5.40i)T+(35.5+61.4i)T2 1 + (9.36 + 5.40i)T + (35.5 + 61.4i)T^{2}
73 1−3.46iT−73T2 1 - 3.46iT - 73T^{2}
79 1−12.4T+79T2 1 - 12.4T + 79T^{2}
83 1−6.92T+83T2 1 - 6.92T + 83T^{2}
89 1+(−7.5+4.33i)T+(44.5−77.0i)T2 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2}
97 1+(10.5+6.06i)T+(48.5+84.0i)T2 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)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

−10.08184081410043997493743685798, −9.441044472061838420289932047170, −8.319228401403583064302418495719, −7.40463126333700147705892906733, −6.40432132134352495390680595765, −5.79999924713761962098103893153, −4.53795606217992001104377864814, −3.73048678992419198325353371622, −2.43662897409828315085239845378, −0.46437949603690812662274263339, 1.31874673548924120852853098846, 3.11270570997036703328696172643, 3.74543383087958232459972163570, 5.44963780273806398433555201717, 6.14415273924661188905352533206, 6.55451431333082299012183313936, 7.88520031332903512421831957486, 8.828249520190240156909224656180, 9.464698692939950995251815970769, 10.39855848153417982046192255960

Graph of the ZZ-function along the critical line