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Properties

Label 2-1008-63.16-c1-0-3
Degree 22
Conductor 10081008
Sign −0.764−0.644i-0.764 - 0.644i
Analytic cond. 8.048928.04892
Root an. cond. 2.837062.83706
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  + (−1.73 − 0.0789i)3-s − 0.460·5-s + (−2.25 + 1.38i)7-s + (2.98 + 0.273i)9-s + 3.64·11-s + (0.730 + 1.26i)13-s + (0.796 + 0.0363i)15-s + (−1.86 − 3.23i)17-s + (2.02 − 3.51i)19-s + (4.01 − 2.20i)21-s − 1.13·23-s − 4.78·25-s + (−5.14 − 0.708i)27-s + (−4.48 + 7.77i)29-s + (−0.257 + 0.445i)31-s + ⋯
L(s)  = 1  + (−0.998 − 0.0455i)3-s − 0.205·5-s + (−0.853 + 0.521i)7-s + (0.995 + 0.0910i)9-s + 1.09·11-s + (0.202 + 0.350i)13-s + (0.205 + 0.00938i)15-s + (−0.452 − 0.784i)17-s + (0.465 − 0.805i)19-s + (0.876 − 0.482i)21-s − 0.236·23-s − 0.957·25-s + (−0.990 − 0.136i)27-s + (−0.833 + 1.44i)29-s + (−0.0462 + 0.0800i)31-s + ⋯

Functional equation

Λ(s)=(1008s/2 ΓC(s) L(s)=((−0.764−0.644i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1008s/2 ΓC(s+1/2) L(s)=((−0.764−0.644i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.764 - 0.644i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10081008    =    24⋅32⋅72^{4} \cdot 3^{2} \cdot 7
Sign: −0.764−0.644i-0.764 - 0.644i
Analytic conductor: 8.048928.04892
Root analytic conductor: 2.837062.83706
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1008(961,⋅)\chi_{1008} (961, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1008, ( :1/2), −0.764−0.644i)(2,\ 1008,\ (\ :1/2),\ -0.764 - 0.644i)

Particular Values

L(1)L(1) ≈\approx 0.40063526620.4006352662
L(12)L(\frac12) ≈\approx 0.40063526620.4006352662
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
3 1+(1.73+0.0789i)T 1 + (1.73 + 0.0789i)T
7 1+(2.25−1.38i)T 1 + (2.25 - 1.38i)T
good5 1+0.460T+5T2 1 + 0.460T + 5T^{2}
11 1−3.64T+11T2 1 - 3.64T + 11T^{2}
13 1+(−0.730−1.26i)T+(−6.5+11.2i)T2 1 + (-0.730 - 1.26i)T + (-6.5 + 11.2i)T^{2}
17 1+(1.86+3.23i)T+(−8.5+14.7i)T2 1 + (1.86 + 3.23i)T + (-8.5 + 14.7i)T^{2}
19 1+(−2.02+3.51i)T+(−9.5−16.4i)T2 1 + (-2.02 + 3.51i)T + (-9.5 - 16.4i)T^{2}
23 1+1.13T+23T2 1 + 1.13T + 23T^{2}
29 1+(4.48−7.77i)T+(−14.5−25.1i)T2 1 + (4.48 - 7.77i)T + (-14.5 - 25.1i)T^{2}
31 1+(0.257−0.445i)T+(−15.5−26.8i)T2 1 + (0.257 - 0.445i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.55−7.88i)T+(−18.5−32.0i)T2 1 + (4.55 - 7.88i)T + (-18.5 - 32.0i)T^{2}
41 1+(0.472+0.819i)T+(−20.5+35.5i)T2 1 + (0.472 + 0.819i)T + (-20.5 + 35.5i)T^{2}
43 1+(4.66−8.07i)T+(−21.5−37.2i)T2 1 + (4.66 - 8.07i)T + (-21.5 - 37.2i)T^{2}
47 1+(−1.16−2.01i)T+(−23.5+40.7i)T2 1 + (-1.16 - 2.01i)T + (-23.5 + 40.7i)T^{2}
53 1+(−6.21−10.7i)T+(−26.5+45.8i)T2 1 + (-6.21 - 10.7i)T + (-26.5 + 45.8i)T^{2}
59 1+(6.44−11.1i)T+(−29.5−51.0i)T2 1 + (6.44 - 11.1i)T + (-29.5 - 51.0i)T^{2}
61 1+(6.04+10.4i)T+(−30.5+52.8i)T2 1 + (6.04 + 10.4i)T + (-30.5 + 52.8i)T^{2}
67 1+(1.16−2.00i)T+(−33.5−58.0i)T2 1 + (1.16 - 2.00i)T + (-33.5 - 58.0i)T^{2}
71 1+1.67T+71T2 1 + 1.67T + 71T^{2}
73 1+(6.62+11.4i)T+(−36.5+63.2i)T2 1 + (6.62 + 11.4i)T + (-36.5 + 63.2i)T^{2}
79 1+(2.50+4.33i)T+(−39.5+68.4i)T2 1 + (2.50 + 4.33i)T + (-39.5 + 68.4i)T^{2}
83 1+(3.32−5.75i)T+(−41.5−71.8i)T2 1 + (3.32 - 5.75i)T + (-41.5 - 71.8i)T^{2}
89 1+(1.36−2.36i)T+(−44.5−77.0i)T2 1 + (1.36 - 2.36i)T + (-44.5 - 77.0i)T^{2}
97 1+(5.59−9.68i)T+(−48.5−84.0i)T2 1 + (5.59 - 9.68i)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

−10.31459945233086739782356200622, −9.359941409415027155559225842119, −8.984125117556910209550034391000, −7.49591318772300041347099260095, −6.74565765151101268822082364419, −6.15004319458635247703476331284, −5.15905943986277968099351958805, −4.20748437566498736808595903149, −3.10029542898103028857316454742, −1.46888033376599572451699296229, 0.21725722712362141910799107253, 1.73673665905131937270034253148, 3.77606647391854451661155897704, 4.01518092750174708105673252263, 5.55950860449618476541696515471, 6.16541151867950841204177029123, 6.96483344040998626347428211401, 7.77962536111066591537894230726, 8.969586908571796919051900080132, 9.905694074114242719893325037915

Graph of the ZZ-function along the critical line