↑

Properties

Label 2-800-20.3-c1-0-3
Degree 22
Conductor 800800
Sign −0.525−0.850i-0.525 - 0.850i
Analytic cond. 6.388036.38803
Root an. cond. 2.527452.52745
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  + (−1 + i)3-s + (3 + 3i)7-s + i·9-s + 2i·11-s + (−3 − 3i)13-s + (−1 + i)17-s + 4·19-s − 6·21-s + (1 − i)23-s + (−4 − 4i)27-s + 10i·31-s + (−2 − 2i)33-s + (1 − i)37-s + 6·39-s − 10·41-s + ⋯
L(s)  = 1  + (−0.577 + 0.577i)3-s + (1.13 + 1.13i)7-s + 0.333i·9-s + 0.603i·11-s + (−0.832 − 0.832i)13-s + (−0.242 + 0.242i)17-s + 0.917·19-s − 1.30·21-s + (0.208 − 0.208i)23-s + (−0.769 − 0.769i)27-s + 1.79i·31-s + (−0.348 − 0.348i)33-s + (0.164 − 0.164i)37-s + 0.960·39-s − 1.56·41-s + ⋯

Functional equation

Λ(s)=(800s/2 ΓC(s) L(s)=((−0.525−0.850i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(800s/2 ΓC(s+1/2) L(s)=((−0.525−0.850i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.525 - 0.850i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 800800    =    25⋅522^{5} \cdot 5^{2}
Sign: −0.525−0.850i-0.525 - 0.850i
Analytic conductor: 6.388036.38803
Root analytic conductor: 2.527452.52745
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ800(543,⋅)\chi_{800} (543, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 800, ( :1/2), −0.525−0.850i)(2,\ 800,\ (\ :1/2),\ -0.525 - 0.850i)

Particular Values

L(1)L(1) ≈\approx 0.573689+1.02897i0.573689 + 1.02897i
L(12)L(\frac12) ≈\approx 0.573689+1.02897i0.573689 + 1.02897i
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
5 1 1
good3 1+(1−i)T−3iT2 1 + (1 - i)T - 3iT^{2}
7 1+(−3−3i)T+7iT2 1 + (-3 - 3i)T + 7iT^{2}
11 1−2iT−11T2 1 - 2iT - 11T^{2}
13 1+(3+3i)T+13iT2 1 + (3 + 3i)T + 13iT^{2}
17 1+(1−i)T−17iT2 1 + (1 - i)T - 17iT^{2}
19 1−4T+19T2 1 - 4T + 19T^{2}
23 1+(−1+i)T−23iT2 1 + (-1 + i)T - 23iT^{2}
29 1−29T2 1 - 29T^{2}
31 1−10iT−31T2 1 - 10iT - 31T^{2}
37 1+(−1+i)T−37iT2 1 + (-1 + i)T - 37iT^{2}
41 1+10T+41T2 1 + 10T + 41T^{2}
43 1+(5−5i)T−43iT2 1 + (5 - 5i)T - 43iT^{2}
47 1+(−3−3i)T+47iT2 1 + (-3 - 3i)T + 47iT^{2}
53 1+(−5−5i)T+53iT2 1 + (-5 - 5i)T + 53iT^{2}
59 1+12T+59T2 1 + 12T + 59T^{2}
61 1−2T+61T2 1 - 2T + 61T^{2}
67 1+(−1−i)T+67iT2 1 + (-1 - i)T + 67iT^{2}
71 1−2iT−71T2 1 - 2iT - 71T^{2}
73 1+(1+i)T+73iT2 1 + (1 + i)T + 73iT^{2}
79 1−8T+79T2 1 - 8T + 79T^{2}
83 1+(5−5i)T−83iT2 1 + (5 - 5i)T - 83iT^{2}
89 1+16iT−89T2 1 + 16iT - 89T^{2}
97 1+(−3+3i)T−97iT2 1 + (-3 + 3i)T - 97iT^{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

−10.53801898130760231602550792017, −9.881633541073524618857510399796, −8.845966807037717208827539248878, −8.055300510133531601149933734628, −7.20776009761738679295576243310, −5.84321312207294119439341714655, −5.04279350927061140336413948223, −4.70724086947760161543118487461, −2.98583320373619828613211231407, −1.78704947084357478927538519404, 0.64506924347125637965437535748, 1.86629275310895654054021106238, 3.58560007532189503648850603494, 4.63355241801052051179115717180, 5.51986834619945442717582828569, 6.68522468942190639094060403261, 7.29577776853698128231862495224, 8.043369381561794008056577988471, 9.177523187138729099815204687783, 10.04601612046077669039541069461

Graph of the ZZ-function along the critical line