↑

Properties

Label 2-980-140.139-c1-0-108
Degree 22
Conductor 980980
Sign −0.155−0.987i-0.155 - 0.987i
Analytic cond. 7.825337.82533
Root an. cond. 2.797382.79738
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  + (0.255 − 1.39i)2-s − 3.18i·3-s + (−1.86 − 0.711i)4-s + (1.80 + 1.31i)5-s + (−4.43 − 0.815i)6-s + (−1.46 + 2.41i)8-s − 7.15·9-s + (2.29 − 2.17i)10-s − 4.51i·11-s + (−2.26 + 5.95i)12-s − 2.22·13-s + (4.19 − 5.75i)15-s + (2.98 + 2.66i)16-s − 2.52·17-s + (−1.83 + 9.94i)18-s − 5.21·19-s + ⋯
L(s)  = 1  + (0.180 − 0.983i)2-s − 1.83i·3-s + (−0.934 − 0.355i)4-s + (0.808 + 0.588i)5-s + (−1.80 − 0.332i)6-s + (−0.519 + 0.854i)8-s − 2.38·9-s + (0.725 − 0.688i)10-s − 1.36i·11-s + (−0.654 + 1.71i)12-s − 0.617·13-s + (1.08 − 1.48i)15-s + (0.746 + 0.665i)16-s − 0.613·17-s + (−0.431 + 2.34i)18-s − 1.19·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.155−0.987i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.155−0.987i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.155 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.155−0.987i-0.155 - 0.987i
Analytic conductor: 7.825337.82533
Root analytic conductor: 2.797382.79738
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ980(979,⋅)\chi_{980} (979, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 980, ( :1/2), −0.155−0.987i)(2,\ 980,\ (\ :1/2),\ -0.155 - 0.987i)

Particular Values

L(1)L(1) ≈\approx 0.648918+0.758693i0.648918 + 0.758693i
L(12)L(\frac12) ≈\approx 0.648918+0.758693i0.648918 + 0.758693i
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+(−0.255+1.39i)T 1 + (-0.255 + 1.39i)T
5 1+(−1.80−1.31i)T 1 + (-1.80 - 1.31i)T
7 1 1
good3 1+3.18iT−3T2 1 + 3.18iT - 3T^{2}
11 1+4.51iT−11T2 1 + 4.51iT - 11T^{2}
13 1+2.22T+13T2 1 + 2.22T + 13T^{2}
17 1+2.52T+17T2 1 + 2.52T + 17T^{2}
19 1+5.21T+19T2 1 + 5.21T + 19T^{2}
23 1−1.71T+23T2 1 - 1.71T + 23T^{2}
29 1+2.31T+29T2 1 + 2.31T + 29T^{2}
31 1+4.62T+31T2 1 + 4.62T + 31T^{2}
37 1+0.336iT−37T2 1 + 0.336iT - 37T^{2}
41 1+3.28iT−41T2 1 + 3.28iT - 41T^{2}
43 1−6.66T+43T2 1 - 6.66T + 43T^{2}
47 1+1.44iT−47T2 1 + 1.44iT - 47T^{2}
53 1+10.0iT−53T2 1 + 10.0iT - 53T^{2}
59 1+3.20T+59T2 1 + 3.20T + 59T^{2}
61 1+6.05iT−61T2 1 + 6.05iT - 61T^{2}
67 1−11.1T+67T2 1 - 11.1T + 67T^{2}
71 1+9.15iT−71T2 1 + 9.15iT - 71T^{2}
73 1−3.24T+73T2 1 - 3.24T + 73T^{2}
79 1+14.2iT−79T2 1 + 14.2iT - 79T^{2}
83 1+11.3iT−83T2 1 + 11.3iT - 83T^{2}
89 1−15.2iT−89T2 1 - 15.2iT - 89T^{2}
97 1+4.49T+97T2 1 + 4.49T + 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

−9.278736896530875386739848773831, −8.664143010711440617057234765667, −7.74688206603551698841335438905, −6.67403524204306313136137790191, −6.05714521547058172142437950251, −5.23857624772435756294851521611, −3.47286731397915693526110946999, −2.46709876806403161228125907663, −1.83819564120811808245754675303, −0.40898582173424791611880539562, 2.49893633576942021668945475610, 4.06374006545645507335392055779, 4.56376696279378333429290940187, 5.24142213815729187663896258548, 6.05635951242905201420134619133, 7.15143083609230036893788406621, 8.390003849947435856585214283546, 9.110113077266661088353107699133, 9.601605646161050447307661708190, 10.19289131099974402363235861406

Graph of the ZZ-function along the critical line