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Properties

Label 2-90e2-5.4-c1-0-57
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.732i·7-s + 1.73·11-s + 1.46i·13-s + 1.26i·17-s − 2.46·19-s − 3.46i·23-s − 4.26·29-s − 7.92·31-s + 4.19i·37-s + 0.803·41-s − 6.73i·43-s + 4.73i·47-s + 6.46·49-s − 10.7i·53-s − 4.26·59-s + ⋯
L(s)  = 1  + 0.276i·7-s + 0.522·11-s + 0.406i·13-s + 0.307i·17-s − 0.565·19-s − 0.722i·23-s − 0.792·29-s − 1.42·31-s + 0.689i·37-s + 0.125·41-s − 1.02i·43-s + 0.690i·47-s + 0.923·49-s − 1.47i·53-s − 0.555·59-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 0.80431690400.8043169040
L(12)L(\frac12) ≈\approx 0.80431690400.8043169040
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 1−0.732iT−7T2 1 - 0.732iT - 7T^{2}
11 1−1.73T+11T2 1 - 1.73T + 11T^{2}
13 1−1.46iT−13T2 1 - 1.46iT - 13T^{2}
17 1−1.26iT−17T2 1 - 1.26iT - 17T^{2}
19 1+2.46T+19T2 1 + 2.46T + 19T^{2}
23 1+3.46iT−23T2 1 + 3.46iT - 23T^{2}
29 1+4.26T+29T2 1 + 4.26T + 29T^{2}
31 1+7.92T+31T2 1 + 7.92T + 31T^{2}
37 1−4.19iT−37T2 1 - 4.19iT - 37T^{2}
41 1−0.803T+41T2 1 - 0.803T + 41T^{2}
43 1+6.73iT−43T2 1 + 6.73iT - 43T^{2}
47 1−4.73iT−47T2 1 - 4.73iT - 47T^{2}
53 1+10.7iT−53T2 1 + 10.7iT - 53T^{2}
59 1+4.26T+59T2 1 + 4.26T + 59T^{2}
61 1+4T+61T2 1 + 4T + 61T^{2}
67 1+14.3iT−67T2 1 + 14.3iT - 67T^{2}
71 1−0.803T+71T2 1 - 0.803T + 71T^{2}
73 1+10.1iT−73T2 1 + 10.1iT - 73T^{2}
79 1+6.39T+79T2 1 + 6.39T + 79T^{2}
83 1−9.12iT−83T2 1 - 9.12iT - 83T^{2}
89 1+5.19T+89T2 1 + 5.19T + 89T^{2}
97 1+2.73iT−97T2 1 + 2.73iT - 97T^{2}
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   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

−7.58532059220180528423111794867, −6.85319282589035744324347460079, −6.26707533240463514297015927530, −5.57102585267548162113678575558, −4.76016505369932647820947080055, −4.00897604053955738836722710170, −3.34179194150733243741301246630, −2.25194070438566794164048200024, −1.59060936046664083177389947721, −0.19290917447958255108560666359, 1.08561613217301888949976881122, 2.02333429146801692597178282020, 2.99951132303552983284904893186, 3.83965080749752124721595933566, 4.38056870104588391142390359373, 5.43373247788282536580030267928, 5.84684302046422547312797541915, 6.76843687147166538265192234201, 7.40601667554485144387382239761, 7.86419007211873361848351230756

Graph of the ZZ-function along the critical line