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Properties

Label 2-90e2-1.1-c1-0-32
Degree 22
Conductor 81008100
Sign 11
Analytic cond. 64.678864.6788
Root an. cond. 8.042318.04231
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 4·13-s + 6·17-s + 2·19-s + 3·23-s + 3·29-s − 10·31-s + 10·37-s + 9·41-s + 4·43-s − 9·47-s − 6·49-s + 6·53-s − 6·59-s − 61-s − 11·67-s + 12·71-s + 4·73-s − 10·79-s + 9·83-s + 9·89-s + 4·91-s + 10·97-s − 18·101-s − 8·103-s − 9·107-s − 19·109-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s + 1.45·17-s + 0.458·19-s + 0.625·23-s + 0.557·29-s − 1.79·31-s + 1.64·37-s + 1.40·41-s + 0.609·43-s − 1.31·47-s − 6/7·49-s + 0.824·53-s − 0.781·59-s − 0.128·61-s − 1.34·67-s + 1.42·71-s + 0.468·73-s − 1.12·79-s + 0.987·83-s + 0.953·89-s + 0.419·91-s + 1.01·97-s − 1.79·101-s − 0.788·103-s − 0.870·107-s − 1.81·109-s + ⋯

Functional equation

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

Invariants

Degree: 22
Conductor: 81008100    =    22⋅34⋅522^{2} \cdot 3^{4} \cdot 5^{2}
Sign: 11
Analytic conductor: 64.678864.6788
Root analytic conductor: 8.042318.04231
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 8100, ( :1/2), 1)(2,\ 8100,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 2.6200462472.620046247
L(12)L(\frac12) ≈\approx 2.6200462472.620046247
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−T+pT2 1 - T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1−4T+pT2 1 - 4 T + p T^{2}
17 1−6T+pT2 1 - 6 T + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−3T+pT2 1 - 3 T + p T^{2}
29 1−3T+pT2 1 - 3 T + p T^{2}
31 1+10T+pT2 1 + 10 T + p T^{2}
37 1−10T+pT2 1 - 10 T + p T^{2}
41 1−9T+pT2 1 - 9 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1+9T+pT2 1 + 9 T + p T^{2}
53 1−6T+pT2 1 - 6 T + p T^{2}
59 1+6T+pT2 1 + 6 T + p T^{2}
61 1+T+pT2 1 + T + p T^{2}
67 1+11T+pT2 1 + 11 T + p T^{2}
71 1−12T+pT2 1 - 12 T + p T^{2}
73 1−4T+pT2 1 - 4 T + p T^{2}
79 1+10T+pT2 1 + 10 T + p T^{2}
83 1−9T+pT2 1 - 9 T + p T^{2}
89 1−9T+pT2 1 - 9 T + p T^{2}
97 1−10T+pT2 1 - 10 T + p T^{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

−7.85183412692991280268209321471, −7.27216683550176494467412774295, −6.34434458519697831088386530205, −5.73985647429667001325056621147, −5.12887363326197211119950499689, −4.22921397807776347434098543844, −3.49051681868049102661843268989, −2.78682666271959125150566575636, −1.59168497822819124815584438671, −0.873023944465869602770010516044, 0.873023944465869602770010516044, 1.59168497822819124815584438671, 2.78682666271959125150566575636, 3.49051681868049102661843268989, 4.22921397807776347434098543844, 5.12887363326197211119950499689, 5.73985647429667001325056621147, 6.34434458519697831088386530205, 7.27216683550176494467412774295, 7.85183412692991280268209321471

Graph of the ZZ-function along the critical line