↑

Properties

Label 2-980-140.139-c1-0-96
Degree 22
Conductor 980980
Sign −0.206+0.978i-0.206 + 0.978i
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  + (1.14 − 0.826i)2-s + 1.52i·3-s + (0.632 − 1.89i)4-s + (0.0967 − 2.23i)5-s + (1.25 + 1.74i)6-s + (−0.843 − 2.69i)8-s + 0.679·9-s + (−1.73 − 2.64i)10-s − 4.56i·11-s + (2.89 + 0.963i)12-s − 2.19·13-s + (3.40 + 0.147i)15-s + (−3.19 − 2.40i)16-s − 6.22·17-s + (0.779 − 0.561i)18-s + 3.83·19-s + ⋯
L(s)  = 1  + (0.811 − 0.584i)2-s + 0.879i·3-s + (0.316 − 0.948i)4-s + (0.0432 − 0.999i)5-s + (0.514 + 0.713i)6-s + (−0.298 − 0.954i)8-s + 0.226·9-s + (−0.549 − 0.835i)10-s − 1.37i·11-s + (0.834 + 0.278i)12-s − 0.608·13-s + (0.878 + 0.0380i)15-s + (−0.799 − 0.600i)16-s − 1.50·17-s + (0.183 − 0.132i)18-s + 0.879·19-s + ⋯

Functional equation

Λ(s)=(980s/2 ΓC(s) L(s)=((−0.206+0.978i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(980s/2 ΓC(s+1/2) L(s)=((−0.206+0.978i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 980 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.206 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 980980    =    22⋅5⋅722^{2} \cdot 5 \cdot 7^{2}
Sign: −0.206+0.978i-0.206 + 0.978i
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), −0.206+0.978i)(2,\ 980,\ (\ :1/2),\ -0.206 + 0.978i)

Particular Values

L(1)L(1) ≈\approx 1.47069−1.81329i1.47069 - 1.81329i
L(12)L(\frac12) ≈\approx 1.47069−1.81329i1.47069 - 1.81329i
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+(−1.14+0.826i)T 1 + (-1.14 + 0.826i)T
5 1+(−0.0967+2.23i)T 1 + (-0.0967 + 2.23i)T
7 1 1
good3 1−1.52iT−3T2 1 - 1.52iT - 3T^{2}
11 1+4.56iT−11T2 1 + 4.56iT - 11T^{2}
13 1+2.19T+13T2 1 + 2.19T + 13T^{2}
17 1+6.22T+17T2 1 + 6.22T + 17T^{2}
19 1−3.83T+19T2 1 - 3.83T + 19T^{2}
23 1−0.430T+23T2 1 - 0.430T + 23T^{2}
29 1+0.473T+29T2 1 + 0.473T + 29T^{2}
31 1−7.59T+31T2 1 - 7.59T + 31T^{2}
37 1+8.44iT−37T2 1 + 8.44iT - 37T^{2}
41 1+1.45iT−41T2 1 + 1.45iT - 41T^{2}
43 1−8.58T+43T2 1 - 8.58T + 43T^{2}
47 1+4.48iT−47T2 1 + 4.48iT - 47T^{2}
53 1−9.23iT−53T2 1 - 9.23iT - 53T^{2}
59 1+3.13T+59T2 1 + 3.13T + 59T^{2}
61 1+5.71iT−61T2 1 + 5.71iT - 61T^{2}
67 1−14.9T+67T2 1 - 14.9T + 67T^{2}
71 1−4.57iT−71T2 1 - 4.57iT - 71T^{2}
73 1−12.2T+73T2 1 - 12.2T + 73T^{2}
79 1−6.20iT−79T2 1 - 6.20iT - 79T^{2}
83 1−7.69iT−83T2 1 - 7.69iT - 83T^{2}
89 1−9.32iT−89T2 1 - 9.32iT - 89T^{2}
97 1+9.05T+97T2 1 + 9.05T + 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

−9.707014206296338958331955654332, −9.287328764027595688252145073942, −8.376659648928959368317311451859, −7.04913345054060719718509125250, −5.90258789872149738143769009450, −5.15304697045377402762981261006, −4.40063816118343567452818661534, −3.67219821074116647303507570455, −2.40527950664686895965153624787, −0.825828449723787310694362672260, 2.02134258650452765597460491954, 2.78980971191265311540899910876, 4.18884539763060867832554047917, 4.96372902290262285551229615499, 6.36071163432435031720456734047, 6.77370063842249558271217520304, 7.42540235601784689150584550247, 8.050881579850480061489194841235, 9.422083344440308807724647626924, 10.25044911313293856488680672593

Graph of the ZZ-function along the critical line