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Properties

Label 2-832-104.101-c1-0-1
Degree 22
Conductor 832832
Sign −0.308−0.951i-0.308 - 0.951i
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  + (0.633 − 0.366i)3-s − 1.73·5-s + (−1.73 − i)7-s + (−1.23 + 2.13i)9-s + (1.73 + 3i)11-s + (1.59 − 3.23i)13-s + (−1.09 + 0.633i)15-s + (−3.23 + 5.59i)17-s + (0.633 − 1.09i)19-s − 1.46·21-s + (0.633 + 1.09i)23-s − 2.00·25-s + 4i·27-s + (−5.59 + 3.23i)29-s + 10.1i·31-s + ⋯
L(s)  = 1  + (0.366 − 0.211i)3-s − 0.774·5-s + (−0.654 − 0.377i)7-s + (−0.410 + 0.711i)9-s + (0.522 + 0.904i)11-s + (0.443 − 0.896i)13-s + (−0.283 + 0.163i)15-s + (−0.783 + 1.35i)17-s + (0.145 − 0.251i)19-s − 0.319·21-s + (0.132 + 0.228i)23-s − 0.400·25-s + 0.769i·27-s + (−1.03 + 0.600i)29-s + 1.83i·31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((−0.308−0.951i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((−0.308−0.951i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.308 - 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: −0.308−0.951i-0.308 - 0.951i
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), −0.308−0.951i)(2,\ 832,\ (\ :1/2),\ -0.308 - 0.951i)

Particular Values

L(1)L(1) ≈\approx 0.471656+0.648647i0.471656 + 0.648647i
L(12)L(\frac12) ≈\approx 0.471656+0.648647i0.471656 + 0.648647i
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+(−1.59+3.23i)T 1 + (-1.59 + 3.23i)T
good3 1+(−0.633+0.366i)T+(1.5−2.59i)T2 1 + (-0.633 + 0.366i)T + (1.5 - 2.59i)T^{2}
5 1+1.73T+5T2 1 + 1.73T + 5T^{2}
7 1+(1.73+i)T+(3.5+6.06i)T2 1 + (1.73 + i)T + (3.5 + 6.06i)T^{2}
11 1+(−1.73−3i)T+(−5.5+9.52i)T2 1 + (-1.73 - 3i)T + (-5.5 + 9.52i)T^{2}
17 1+(3.23−5.59i)T+(−8.5−14.7i)T2 1 + (3.23 - 5.59i)T + (-8.5 - 14.7i)T^{2}
19 1+(−0.633+1.09i)T+(−9.5−16.4i)T2 1 + (-0.633 + 1.09i)T + (-9.5 - 16.4i)T^{2}
23 1+(−0.633−1.09i)T+(−11.5+19.9i)T2 1 + (-0.633 - 1.09i)T + (-11.5 + 19.9i)T^{2}
29 1+(5.59−3.23i)T+(14.5−25.1i)T2 1 + (5.59 - 3.23i)T + (14.5 - 25.1i)T^{2}
31 1−10.1iT−31T2 1 - 10.1iT - 31T^{2}
37 1+(0.598+1.03i)T+(−18.5+32.0i)T2 1 + (0.598 + 1.03i)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+(−7.26−4.19i)T+(21.5+37.2i)T2 1 + (-7.26 - 4.19i)T + (21.5 + 37.2i)T^{2}
47 1−5.66iT−47T2 1 - 5.66iT - 47T^{2}
53 1−5.53iT−53T2 1 - 5.53iT - 53T^{2}
59 1+(−6.46+11.1i)T+(−29.5−51.0i)T2 1 + (-6.46 + 11.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.40+3.69i)T+(30.5+52.8i)T2 1 + (6.40 + 3.69i)T + (30.5 + 52.8i)T^{2}
67 1+(1.09+1.90i)T+(−33.5+58.0i)T2 1 + (1.09 + 1.90i)T + (-33.5 + 58.0i)T^{2}
71 1+(4.90+2.83i)T+(35.5+61.4i)T2 1 + (4.90 + 2.83i)T + (35.5 + 61.4i)T^{2}
73 1−5.19iT−73T2 1 - 5.19iT - 73T^{2}
79 1−6T+79T2 1 - 6T + 79T^{2}
83 1−9.46T+83T2 1 - 9.46T + 83T^{2}
89 1+(−7.39+4.26i)T+(44.5−77.0i)T2 1 + (-7.39 + 4.26i)T + (44.5 - 77.0i)T^{2}
97 1+(5.19+3i)T+(48.5+84.0i)T2 1 + (5.19 + 3i)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

−10.65679494974465807912135371129, −9.557680948120433187923311349020, −8.653684458413111282955743035154, −7.913715352367541783703451502710, −7.17247970340504704018879209695, −6.27071918000224318242778360308, −5.06224650750426655402974420933, −3.94786455650009749135626388239, −3.14921534986322303045871277928, −1.68689780505711253617367394809, 0.36881444338327829491246210731, 2.47934172792982154625167041294, 3.61045688886475114760253619558, 4.16366816529842385055638701453, 5.71267156591568723983466626995, 6.45633425979588760882457860160, 7.39893191148185263326444071232, 8.476254573469116374815935146526, 9.169579113922946061716344237283, 9.589136774981603846218848272997

Graph of the ZZ-function along the critical line