↑

Properties

Label 2-832-208.181-c1-0-10
Degree 22
Conductor 832832
Sign 0.0103−0.999i0.0103 - 0.999i
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−2.15 + 2.15i)3-s + (0.707 − 0.707i)5-s − 0.223·7-s − 6.31i·9-s + (1.41 − 1.41i)11-s + (1.34 + 3.34i)13-s + 3.05i·15-s + 3·17-s + (4.46 + 4.46i)19-s + (0.483 − 0.483i)21-s − 6.31i·23-s + 4i·25-s + (7.15 + 7.15i)27-s + (0.316 − 0.316i)29-s − 4.24i·31-s + ⋯
L(s)  = 1  + (−1.24 + 1.24i)3-s + (0.316 − 0.316i)5-s − 0.0846·7-s − 2.10i·9-s + (0.426 − 0.426i)11-s + (0.373 + 0.927i)13-s + 0.788i·15-s + 0.727·17-s + (1.02 + 1.02i)19-s + (0.105 − 0.105i)21-s − 1.31i·23-s + 0.800i·25-s + (1.37 + 1.37i)27-s + (0.0587 − 0.0587i)29-s − 0.762i·31-s + ⋯

Functional equation

Λ(s)=(832s/2 ΓC(s) L(s)=((0.0103−0.999i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(832s/2 ΓC(s+1/2) L(s)=((0.0103−0.999i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 832 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0103 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 832832    =    26⋅132^{6} \cdot 13
Sign: 0.0103−0.999i0.0103 - 0.999i
Analytic conductor: 6.643556.64355
Root analytic conductor: 2.577502.57750
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ832(753,⋅)\chi_{832} (753, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 832, ( :1/2), 0.0103−0.999i)(2,\ 832,\ (\ :1/2),\ 0.0103 - 0.999i)

Particular Values

L(1)L(1) ≈\approx 0.744734+0.737060i0.744734 + 0.737060i
L(12)L(\frac12) ≈\approx 0.744734+0.737060i0.744734 + 0.737060i
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+(−1.34−3.34i)T 1 + (-1.34 - 3.34i)T
good3 1+(2.15−2.15i)T−3iT2 1 + (2.15 - 2.15i)T - 3iT^{2}
5 1+(−0.707+0.707i)T−5iT2 1 + (-0.707 + 0.707i)T - 5iT^{2}
7 1+0.223T+7T2 1 + 0.223T + 7T^{2}
11 1+(−1.41+1.41i)T−11iT2 1 + (-1.41 + 1.41i)T - 11iT^{2}
17 1−3T+17T2 1 - 3T + 17T^{2}
19 1+(−4.46−4.46i)T+19iT2 1 + (-4.46 - 4.46i)T + 19iT^{2}
23 1+6.31iT−23T2 1 + 6.31iT - 23T^{2}
29 1+(−0.316+0.316i)T−29iT2 1 + (-0.316 + 0.316i)T - 29iT^{2}
31 1+4.24iT−31T2 1 + 4.24iT - 31T^{2}
37 1+(6.58−6.58i)T−37iT2 1 + (6.58 - 6.58i)T - 37iT^{2}
41 1+41T2 1 + 41T^{2}
43 1+(−5.47−5.47i)T+43iT2 1 + (-5.47 - 5.47i)T + 43iT^{2}
47 1−10.5iT−47T2 1 - 10.5iT - 47T^{2}
53 1+(−3.63−3.63i)T+53iT2 1 + (-3.63 - 3.63i)T + 53iT^{2}
59 1+(7.74−7.74i)T−59iT2 1 + (7.74 - 7.74i)T - 59iT^{2}
61 1+(−4+4i)T−61iT2 1 + (-4 + 4i)T - 61iT^{2}
67 1+(4.69+4.69i)T+67iT2 1 + (4.69 + 4.69i)T + 67iT^{2}
71 1−0.671T+71T2 1 - 0.671T + 71T^{2}
73 1−3.79T+73T2 1 - 3.79T + 73T^{2}
79 1−10.9T+79T2 1 - 10.9T + 79T^{2}
83 1+(−11.9−11.9i)T+83iT2 1 + (-11.9 - 11.9i)T + 83iT^{2}
89 1−14.0T+89T2 1 - 14.0T + 89T^{2}
97 1−8.48iT−97T2 1 - 8.48iT - 97T^{2}
show more
show less
   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

−10.48487221266454715320843876303, −9.556358020454650531507958397927, −9.221523746296570055027980412377, −7.957108664103937307117943761351, −6.53429905413934967786480192266, −5.95115713206359878993024502752, −5.11161159052904802594091141464, −4.25490053697503511689815446507, −3.34511545692806006211156433979, −1.19140353993496779957767347475, 0.73882947308169173288339686632, 1.94260315382173235131816008347, 3.38651929316354210598924308379, 5.14100053863742282645504772952, 5.61578504518669907092463270300, 6.61138198535383901310627246771, 7.21892560064785974859828155381, 7.953729110881248414097938918486, 9.220017351223195023747983694452, 10.29324062909346193378940504547

Graph of the ZZ-function along the critical line