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Properties

Label 2-5376-1.1-c1-0-12
Degree 22
Conductor 53765376
Sign 11
Analytic cond. 42.927542.9275
Root an. cond. 6.551916.55191
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 1.29·5-s − 7-s + 9-s − 5.01·11-s + 3.71·13-s − 1.29·15-s − 4.41·17-s − 7.71·19-s + 21-s + 2.41·23-s − 3.31·25-s − 27-s − 7.71·29-s − 3.71·31-s + 5.01·33-s − 1.29·35-s + 10.3·37-s − 3.71·39-s + 11.0·41-s + 2.59·43-s + 1.29·45-s + 10.0·47-s + 49-s + 4.41·51-s + 10.3·53-s − 6.51·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 0.581·5-s − 0.377·7-s + 0.333·9-s − 1.51·11-s + 1.02·13-s − 0.335·15-s − 1.07·17-s − 1.76·19-s + 0.218·21-s + 0.502·23-s − 0.662·25-s − 0.192·27-s − 1.43·29-s − 0.666·31-s + 0.872·33-s − 0.219·35-s + 1.69·37-s − 0.594·39-s + 1.71·41-s + 0.396·43-s + 0.193·45-s + 1.46·47-s + 0.142·49-s + 0.617·51-s + 1.41·53-s − 0.878·55-s + ⋯

Functional equation

Λ(s)=(5376s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(5376s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 5376 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 53765376    =    28⋅3⋅72^{8} \cdot 3 \cdot 7
Sign: 11
Analytic conductor: 42.927542.9275
Root analytic conductor: 6.551916.55191
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 5376, ( :1/2), 1)(2,\ 5376,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.1208433281.120843328
L(12)L(\frac12) ≈\approx 1.1208433281.120843328
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+T 1 + T
7 1+T 1 + T
good5 1−1.29T+5T2 1 - 1.29T + 5T^{2}
11 1+5.01T+11T2 1 + 5.01T + 11T^{2}
13 1−3.71T+13T2 1 - 3.71T + 13T^{2}
17 1+4.41T+17T2 1 + 4.41T + 17T^{2}
19 1+7.71T+19T2 1 + 7.71T + 19T^{2}
23 1−2.41T+23T2 1 - 2.41T + 23T^{2}
29 1+7.71T+29T2 1 + 7.71T + 29T^{2}
31 1+3.71T+31T2 1 + 3.71T + 31T^{2}
37 1−10.3T+37T2 1 - 10.3T + 37T^{2}
41 1−11.0T+41T2 1 - 11.0T + 41T^{2}
43 1−2.59T+43T2 1 - 2.59T + 43T^{2}
47 1−10.0T+47T2 1 - 10.0T + 47T^{2}
53 1−10.3T+53T2 1 - 10.3T + 53T^{2}
59 1−8.82T+59T2 1 - 8.82T + 59T^{2}
61 1+9.11T+61T2 1 + 9.11T + 61T^{2}
67 1+8T+67T2 1 + 8T + 67T^{2}
71 1+7.61T+71T2 1 + 7.61T + 71T^{2}
73 1−12.5T+73T2 1 - 12.5T + 73T^{2}
79 1−10.5T+79T2 1 - 10.5T + 79T^{2}
83 1−9.19T+83T2 1 - 9.19T + 83T^{2}
89 1−4.98T+89T2 1 - 4.98T + 89T^{2}
97 1+12.0T+97T2 1 + 12.0T + 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

−8.120414342847235928699442399508, −7.43532671078756032548172752359, −6.60089970207422828455165620538, −5.82965249470624082967503989763, −5.63785801483813711219700072779, −4.44884574885318322983372522256, −3.90980681963584909931578882811, −2.57088737417535729235426554899, −2.04036864979100301169130415702, −0.55983061597845957113495575515, 0.55983061597845957113495575515, 2.04036864979100301169130415702, 2.57088737417535729235426554899, 3.90980681963584909931578882811, 4.44884574885318322983372522256, 5.63785801483813711219700072779, 5.82965249470624082967503989763, 6.60089970207422828455165620538, 7.43532671078756032548172752359, 8.120414342847235928699442399508

Graph of the ZZ-function along the critical line