↑

Properties

Label 2-10e3-8.5-c1-0-35
Degree 22
Conductor 10001000
Sign −0.480−0.877i-0.480 - 0.877i
Analytic cond. 7.985047.98504
Root an. cond. 2.825782.82578
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  + (−0.493 + 1.32i)2-s + 1.83i·3-s + (−1.51 − 1.30i)4-s + (−2.43 − 0.906i)6-s + 1.31·7-s + (2.48 − 1.35i)8-s − 0.368·9-s − 6.60i·11-s + (2.40 − 2.77i)12-s + 2.96i·13-s + (−0.649 + 1.74i)14-s + (0.574 + 3.95i)16-s + 2.69·17-s + (0.181 − 0.487i)18-s + 4.97i·19-s + ⋯
L(s)  = 1  + (−0.349 + 0.937i)2-s + 1.05i·3-s + (−0.756 − 0.654i)4-s + (−0.992 − 0.369i)6-s + 0.497·7-s + (0.877 − 0.480i)8-s − 0.122·9-s − 1.99i·11-s + (0.693 − 0.801i)12-s + 0.822i·13-s + (−0.173 + 0.465i)14-s + (0.143 + 0.989i)16-s + 0.653·17-s + (0.0428 − 0.115i)18-s + 1.14i·19-s + ⋯

Functional equation

Λ(s)=(1000s/2 ΓC(s) L(s)=((−0.480−0.877i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(1000s/2 ΓC(s+1/2) L(s)=((−0.480−0.877i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1000 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.480 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10001000    =    23⋅532^{3} \cdot 5^{3}
Sign: −0.480−0.877i-0.480 - 0.877i
Analytic conductor: 7.985047.98504
Root analytic conductor: 2.825782.82578
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ1000(501,⋅)\chi_{1000} (501, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 1000, ( :1/2), −0.480−0.877i)(2,\ 1000,\ (\ :1/2),\ -0.480 - 0.877i)

Particular Values

L(1)L(1) ≈\approx 0.705323+1.19017i0.705323 + 1.19017i
L(12)L(\frac12) ≈\approx 0.705323+1.19017i0.705323 + 1.19017i
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+(0.493−1.32i)T 1 + (0.493 - 1.32i)T
5 1 1
good3 1−1.83iT−3T2 1 - 1.83iT - 3T^{2}
7 1−1.31T+7T2 1 - 1.31T + 7T^{2}
11 1+6.60iT−11T2 1 + 6.60iT - 11T^{2}
13 1−2.96iT−13T2 1 - 2.96iT - 13T^{2}
17 1−2.69T+17T2 1 - 2.69T + 17T^{2}
19 1−4.97iT−19T2 1 - 4.97iT - 19T^{2}
23 1−3.73T+23T2 1 - 3.73T + 23T^{2}
29 1−5.52iT−29T2 1 - 5.52iT - 29T^{2}
31 1−8.36T+31T2 1 - 8.36T + 31T^{2}
37 1+7.19iT−37T2 1 + 7.19iT - 37T^{2}
41 1+3.77T+41T2 1 + 3.77T + 41T^{2}
43 1−3.07iT−43T2 1 - 3.07iT - 43T^{2}
47 1−8.77T+47T2 1 - 8.77T + 47T^{2}
53 1+0.0464iT−53T2 1 + 0.0464iT - 53T^{2}
59 1+1.02iT−59T2 1 + 1.02iT - 59T^{2}
61 1+5.23iT−61T2 1 + 5.23iT - 61T^{2}
67 1−10.8iT−67T2 1 - 10.8iT - 67T^{2}
71 1−9.35T+71T2 1 - 9.35T + 71T^{2}
73 1+12.4T+73T2 1 + 12.4T + 73T^{2}
79 1−1.43T+79T2 1 - 1.43T + 79T^{2}
83 1−13.0iT−83T2 1 - 13.0iT - 83T^{2}
89 1−3.94T+89T2 1 - 3.94T + 89T^{2}
97 1+1.00T+97T2 1 + 1.00T + 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

−10.14324898119898556060600627932, −9.255130801322501229620149797990, −8.622478478863164538978721791775, −7.949933300171837058325147437683, −6.84343555896184440652310384651, −5.85489592940129073995158644527, −5.19540873253243356950715387351, −4.20918072918494792083683624615, −3.35725149956471953933720222072, −1.18830655386051693582383408751, 0.926473497978455344332329221674, 1.97533306534597165650988403871, 2.85932545375616187684892847839, 4.40873630239642598115040954888, 5.03267080122665256444714300666, 6.59483956597848944104481779477, 7.49023506503109278447462851038, 7.87929522339623236113312113785, 8.917273860921809490417133716083, 9.953572475115529677295268125314

Graph of the ZZ-function along the critical line