↑

Properties

Label 2-28e2-112.109-c1-0-18
Degree 22
Conductor 784784
Sign −0.615−0.788i-0.615 - 0.788i
Analytic cond. 6.260276.26027
Root an. cond. 2.502052.50205
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  + (1.36 + 0.366i)2-s + (−1.36 + 0.366i)3-s + (1.73 + i)4-s + (−1.36 − 0.366i)5-s − 2·6-s + (1.99 + 2i)8-s + (−0.866 + 0.5i)9-s + (−1.73 − i)10-s + (0.366 + 1.36i)11-s + (−2.73 − 0.732i)12-s + (1 + i)13-s + 2·15-s + (1.99 + 3.46i)16-s + (−1 + 1.73i)17-s + (−1.36 + 0.366i)18-s + (−1.09 + 4.09i)19-s + ⋯
L(s)  = 1  + (0.965 + 0.258i)2-s + (−0.788 + 0.211i)3-s + (0.866 + 0.5i)4-s + (−0.610 − 0.163i)5-s − 0.816·6-s + (0.707 + 0.707i)8-s + (−0.288 + 0.166i)9-s + (−0.547 − 0.316i)10-s + (0.110 + 0.411i)11-s + (−0.788 − 0.211i)12-s + (0.277 + 0.277i)13-s + 0.516·15-s + (0.499 + 0.866i)16-s + (−0.242 + 0.420i)17-s + (−0.321 + 0.0862i)18-s + (−0.251 + 0.940i)19-s + ⋯

Functional equation

Λ(s)=(784s/2 ΓC(s) L(s)=((−0.615−0.788i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(784s/2 ΓC(s+1/2) L(s)=((−0.615−0.788i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 784 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 784784    =    24⋅722^{4} \cdot 7^{2}
Sign: −0.615−0.788i-0.615 - 0.788i
Analytic conductor: 6.260276.26027
Root analytic conductor: 2.502052.50205
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ784(557,⋅)\chi_{784} (557, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 784, ( :1/2), −0.615−0.788i)(2,\ 784,\ (\ :1/2),\ -0.615 - 0.788i)

Particular Values

L(1)L(1) ≈\approx 0.637806+1.30698i0.637806 + 1.30698i
L(12)L(\frac12) ≈\approx 0.637806+1.30698i0.637806 + 1.30698i
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+(−1.36−0.366i)T 1 + (-1.36 - 0.366i)T
7 1 1
good3 1+(1.36−0.366i)T+(2.59−1.5i)T2 1 + (1.36 - 0.366i)T + (2.59 - 1.5i)T^{2}
5 1+(1.36+0.366i)T+(4.33+2.5i)T2 1 + (1.36 + 0.366i)T + (4.33 + 2.5i)T^{2}
11 1+(−0.366−1.36i)T+(−9.52+5.5i)T2 1 + (-0.366 - 1.36i)T + (-9.52 + 5.5i)T^{2}
13 1+(−1−i)T+13iT2 1 + (-1 - i)T + 13iT^{2}
17 1+(1−1.73i)T+(−8.5−14.7i)T2 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2}
19 1+(1.09−4.09i)T+(−16.4−9.5i)T2 1 + (1.09 - 4.09i)T + (-16.4 - 9.5i)T^{2}
23 1+(5.19−3i)T+(11.5−19.9i)T2 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2}
29 1+(−3−3i)T+29iT2 1 + (-3 - 3i)T + 29iT^{2}
31 1+(4−6.92i)T+(−15.5−26.8i)T2 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2}
37 1+(4.09+1.09i)T+(32.0+18.5i)T2 1 + (4.09 + 1.09i)T + (32.0 + 18.5i)T^{2}
41 1−41T2 1 - 41T^{2}
43 1+(−5+5i)T−43iT2 1 + (-5 + 5i)T - 43iT^{2}
47 1+(−4−6.92i)T+(−23.5+40.7i)T2 1 + (-4 - 6.92i)T + (-23.5 + 40.7i)T^{2}
53 1+(1.83+6.83i)T+(−45.8+26.5i)T2 1 + (1.83 + 6.83i)T + (-45.8 + 26.5i)T^{2}
59 1+(−1.09−4.09i)T+(−51.0+29.5i)T2 1 + (-1.09 - 4.09i)T + (-51.0 + 29.5i)T^{2}
61 1+(−3.29+12.2i)T+(−52.8−30.5i)T2 1 + (-3.29 + 12.2i)T + (-52.8 - 30.5i)T^{2}
67 1+(−6.83+1.83i)T+(58.0−33.5i)T2 1 + (-6.83 + 1.83i)T + (58.0 - 33.5i)T^{2}
71 1−10iT−71T2 1 - 10iT - 71T^{2}
73 1+(−3.46−2i)T+(36.5+63.2i)T2 1 + (-3.46 - 2i)T + (36.5 + 63.2i)T^{2}
79 1+(−39.5+68.4i)T2 1 + (-39.5 + 68.4i)T^{2}
83 1+(−1−i)T+83iT2 1 + (-1 - i)T + 83iT^{2}
89 1+(−3.46+2i)T+(44.5−77.0i)T2 1 + (-3.46 + 2i)T + (44.5 - 77.0i)T^{2}
97 1−2T+97T2 1 - 2T + 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

−10.86526175226527659985543761143, −10.09024142914215116793362636902, −8.613235494106915330104761898068, −7.901618339839388105951051010645, −6.89127760919081632955773886059, −6.01416840664914436388468621447, −5.28190650925256759172563825635, −4.30296248885310855361922526969, −3.54977982277054317555619917832, −1.94805754406740269319251515750, 0.56495862658599632052662303324, 2.42153094949907804958072285165, 3.59611010321157134259589652330, 4.51609685287121078803663308180, 5.60939877429752001293440665617, 6.24045477026544824353755734131, 7.09266228288039239077231175608, 8.069476125818346261938169996748, 9.255716778847940030909001990245, 10.44953683997457855633812806059

Graph of the ZZ-function along the critical line