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Properties

Label 2-832-104.101-c1-0-3
Degree 22
Conductor 832832
Sign 0.494−0.869i0.494 - 0.869i
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  + (−2.82 + 1.63i)3-s − 1.11·5-s + (−3.79 − 2.18i)7-s + (3.81 − 6.61i)9-s + (−1.09 − 1.89i)11-s + (1 + 3.46i)13-s + (3.15 − 1.81i)15-s + (0.942 − 1.63i)17-s + (2.05 − 3.56i)19-s + 14.2·21-s + (2.05 + 3.56i)23-s − 3.75·25-s + 15.1i·27-s + (−8.28 + 4.78i)29-s + 2i·31-s + ⋯
L(s)  = 1  + (−1.63 + 0.941i)3-s − 0.498·5-s + (−1.43 − 0.827i)7-s + (1.27 − 2.20i)9-s + (−0.329 − 0.570i)11-s + (0.277 + 0.960i)13-s + (0.813 − 0.469i)15-s + (0.228 − 0.395i)17-s + (0.472 − 0.818i)19-s + 3.11·21-s + (0.429 + 0.743i)23-s − 0.751·25-s + 2.91i·27-s + (−1.53 + 0.888i)29-s + 0.359i·31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.494−0.869i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.494−0.869i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.494−0.869i0.494 - 0.869i
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), 0.494−0.869i)(2,\ 832,\ (\ :1/2),\ 0.494 - 0.869i)

Particular Values

L(1)L(1) ≈\approx 0.381200+0.221718i0.381200 + 0.221718i
L(12)L(\frac12) ≈\approx 0.381200+0.221718i0.381200 + 0.221718i
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+(−1−3.46i)T 1 + (-1 - 3.46i)T
good3 1+(2.82−1.63i)T+(1.5−2.59i)T2 1 + (2.82 - 1.63i)T + (1.5 - 2.59i)T^{2}
5 1+1.11T+5T2 1 + 1.11T + 5T^{2}
7 1+(3.79+2.18i)T+(3.5+6.06i)T2 1 + (3.79 + 2.18i)T + (3.5 + 6.06i)T^{2}
11 1+(1.09+1.89i)T+(−5.5+9.52i)T2 1 + (1.09 + 1.89i)T + (-5.5 + 9.52i)T^{2}
17 1+(−0.942+1.63i)T+(−8.5−14.7i)T2 1 + (-0.942 + 1.63i)T + (-8.5 - 14.7i)T^{2}
19 1+(−2.05+3.56i)T+(−9.5−16.4i)T2 1 + (-2.05 + 3.56i)T + (-9.5 - 16.4i)T^{2}
23 1+(−2.05−3.56i)T+(−11.5+19.9i)T2 1 + (-2.05 - 3.56i)T + (-11.5 + 19.9i)T^{2}
29 1+(8.28−4.78i)T+(14.5−25.1i)T2 1 + (8.28 - 4.78i)T + (14.5 - 25.1i)T^{2}
31 1−2iT−31T2 1 - 2iT - 31T^{2}
37 1+(1.20+2.08i)T+(−18.5+32.0i)T2 1 + (1.20 + 2.08i)T + (-18.5 + 32.0i)T^{2}
41 1+(−6.17+3.56i)T+(20.5−35.5i)T2 1 + (-6.17 + 3.56i)T + (20.5 - 35.5i)T^{2}
43 1+(−0.892−0.515i)T+(21.5+37.2i)T2 1 + (-0.892 - 0.515i)T + (21.5 + 37.2i)T^{2}
47 1−7.11iT−47T2 1 - 7.11iT - 47T^{2}
53 1+6.30iT−53T2 1 + 6.30iT - 53T^{2}
59 1+(−0.126+0.219i)T+(−29.5−51.0i)T2 1 + (-0.126 + 0.219i)T + (-29.5 - 51.0i)T^{2}
61 1+(1.5+0.866i)T+(30.5+52.8i)T2 1 + (1.5 + 0.866i)T + (30.5 + 52.8i)T^{2}
67 1+(−3.79−6.56i)T+(−33.5+58.0i)T2 1 + (-3.79 - 6.56i)T + (-33.5 + 58.0i)T^{2}
71 1+(−10.4−6.00i)T+(35.5+61.4i)T2 1 + (-10.4 - 6.00i)T + (35.5 + 61.4i)T^{2}
73 1−9.36iT−73T2 1 - 9.36iT - 73T^{2}
79 1−5.90T+79T2 1 - 5.90T + 79T^{2}
83 1−14.7T+83T2 1 - 14.7T + 83T^{2}
89 1+(−8.28+4.78i)T+(44.5−77.0i)T2 1 + (-8.28 + 4.78i)T + (44.5 - 77.0i)T^{2}
97 1+(−1.5−0.866i)T+(48.5+84.0i)T2 1 + (-1.5 - 0.866i)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

−10.52373579668064855591024321359, −9.527381996864986956587130632782, −9.242531427299389420766610158718, −7.38539205015700027484341903611, −6.78978620394751888910468397233, −5.91151333684003944968030966180, −5.08930954486058471182581015713, −3.95616532641507565190306080839, −3.44583207735925992756979815829, −0.69647389123063145062076189388, 0.45814125644956027364944069401, 2.17508179707478980544665167586, 3.64348705212428179153206031237, 5.08506704271235924317696348899, 5.94320143288435730475186022703, 6.29681877004968036746832382115, 7.44891029544189423556271099367, 7.971420297164390696329267727239, 9.455178566554113151241622314938, 10.25519412001410546495349664659

Graph of the ZZ-function along the critical line