↑

Properties

Label 2-90e2-5.4-c1-0-16
Degree 22
Conductor 81008100
Sign −0.447−0.894i-0.447 - 0.894i
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
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  + 2.73i·7-s − 1.73·11-s + 5.46i·13-s − 4.73i·17-s + 4.46·19-s − 3.46i·23-s − 7.73·29-s + 5.92·31-s + 6.19i·37-s + 11.1·41-s + 3.26i·43-s − 1.26i·47-s − 0.464·49-s + 7.26i·53-s − 7.73·59-s + ⋯
L(s)  = 1  + 1.03i·7-s − 0.522·11-s + 1.51i·13-s − 1.14i·17-s + 1.02·19-s − 0.722i·23-s − 1.43·29-s + 1.06·31-s + 1.01i·37-s + 1.74·41-s + 0.498i·43-s − 0.184i·47-s − 0.0663·49-s + 0.998i·53-s − 1.00·59-s + ⋯

Functional equation

Λ(s)=(8100s/2 ΓC(s) L(s)=((−0.447−0.894i) Λ‾(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
Λ(s)=(8100s/2 ΓC(s+1/2) L(s)=((−0.447−0.894i) Λ‾(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 8100 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 81008100    =    22⋅34⋅522^{2} \cdot 3^{4} \cdot 5^{2}
Sign: −0.447−0.894i-0.447 - 0.894i
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: χ8100(649,⋅)\chi_{8100} (649, \cdot )
Primitive: yes
Self-dual: no
Analytic rank: 00
Selberg data: (2, 8100, ( :1/2), −0.447−0.894i)(2,\ 8100,\ (\ :1/2),\ -0.447 - 0.894i)

Particular Values

L(1)L(1) ≈\approx 1.5010431171.501043117
L(12)L(\frac12) ≈\approx 1.5010431171.501043117
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
3 1 1
5 1 1
good7 1−2.73iT−7T2 1 - 2.73iT - 7T^{2}
11 1+1.73T+11T2 1 + 1.73T + 11T^{2}
13 1−5.46iT−13T2 1 - 5.46iT - 13T^{2}
17 1+4.73iT−17T2 1 + 4.73iT - 17T^{2}
19 1−4.46T+19T2 1 - 4.46T + 19T^{2}
23 1+3.46iT−23T2 1 + 3.46iT - 23T^{2}
29 1+7.73T+29T2 1 + 7.73T + 29T^{2}
31 1−5.92T+31T2 1 - 5.92T + 31T^{2}
37 1−6.19iT−37T2 1 - 6.19iT - 37T^{2}
41 1−11.1T+41T2 1 - 11.1T + 41T^{2}
43 1−3.26iT−43T2 1 - 3.26iT - 43T^{2}
47 1+1.26iT−47T2 1 + 1.26iT - 47T^{2}
53 1−7.26iT−53T2 1 - 7.26iT - 53T^{2}
59 1+7.73T+59T2 1 + 7.73T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+6.39iT−67T2 1 + 6.39iT - 67T^{2}
71 1−11.1T+71T2 1 - 11.1T + 71T^{2}
73 1+0.196iT−73T2 1 + 0.196iT - 73T^{2}
79 1−14.3T+79T2 1 - 14.3T + 79T^{2}
83 1−15.1iT−83T2 1 - 15.1iT - 83T^{2}
89 1−5.19T+89T2 1 - 5.19T + 89T^{2}
97 1+0.732iT−97T2 1 + 0.732iT - 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

−7.940954928845129384283285564126, −7.45268437415367521718621233475, −6.57360411317330185410213170591, −6.05088545356677353378753082065, −5.14229331087035426504742197099, −4.72223964476221415109475682988, −3.75579654970022471664609909864, −2.71610249868806172257141623291, −2.28987874419423450495269966416, −1.09441886584919454489348562437, 0.39250421923980820701298727275, 1.29255634186707623467288052427, 2.43565002958516511346958203896, 3.41975297551013825950267292812, 3.85160082984493484948385184900, 4.83742244922653844135233795420, 5.60779582879670041788581747456, 6.04710551684708544151201130935, 7.13073767539361900499695839854, 7.74405087969558783505946599421

Graph of the ZZ-function along the critical line