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Properties

Label 2-5376-1.1-c1-0-28
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 + 2.44·5-s + 7-s + 9-s − 4.44·11-s + 2.89·13-s − 2.44·15-s + 6.44·17-s − 2.89·19-s − 21-s + 0.449·23-s + 0.999·25-s − 27-s + 2.89·29-s + 6.89·31-s + 4.44·33-s + 2.44·35-s + 2·37-s − 2.89·39-s − 7.34·41-s − 3.10·43-s + 2.44·45-s − 0.898·47-s + 49-s − 6.44·51-s + 10·53-s − 10.8·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.09·5-s + 0.377·7-s + 0.333·9-s − 1.34·11-s + 0.804·13-s − 0.632·15-s + 1.56·17-s − 0.665·19-s − 0.218·21-s + 0.0937·23-s + 0.199·25-s − 0.192·27-s + 0.538·29-s + 1.23·31-s + 0.774·33-s + 0.414·35-s + 0.328·37-s − 0.464·39-s − 1.14·41-s − 0.472·43-s + 0.365·45-s − 0.131·47-s + 0.142·49-s − 0.903·51-s + 1.37·53-s − 1.46·55-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 2.1241474102.124147410
L(12)L(\frac12) ≈\approx 2.1241474102.124147410
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 1−2.44T+5T2 1 - 2.44T + 5T^{2}
11 1+4.44T+11T2 1 + 4.44T + 11T^{2}
13 1−2.89T+13T2 1 - 2.89T + 13T^{2}
17 1−6.44T+17T2 1 - 6.44T + 17T^{2}
19 1+2.89T+19T2 1 + 2.89T + 19T^{2}
23 1−0.449T+23T2 1 - 0.449T + 23T^{2}
29 1−2.89T+29T2 1 - 2.89T + 29T^{2}
31 1−6.89T+31T2 1 - 6.89T + 31T^{2}
37 1−2T+37T2 1 - 2T + 37T^{2}
41 1+7.34T+41T2 1 + 7.34T + 41T^{2}
43 1+3.10T+43T2 1 + 3.10T + 43T^{2}
47 1+0.898T+47T2 1 + 0.898T + 47T^{2}
53 1−10T+53T2 1 - 10T + 53T^{2}
59 1+4.89T+59T2 1 + 4.89T + 59T^{2}
61 1+6T+61T2 1 + 6T + 61T^{2}
67 1−9.79T+67T2 1 - 9.79T + 67T^{2}
71 1−9.34T+71T2 1 - 9.34T + 71T^{2}
73 1+10.8T+73T2 1 + 10.8T + 73T^{2}
79 1−12.8T+79T2 1 - 12.8T + 79T^{2}
83 1−5.79T+83T2 1 - 5.79T + 83T^{2}
89 1−7.34T+89T2 1 - 7.34T + 89T^{2}
97 1−5.10T+97T2 1 - 5.10T + 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

−8.125729616692856629127428231228, −7.55007030563806169005130121214, −6.46393892779863554127241669718, −6.03793394555653502998608530406, −5.26105147032306915020766840440, −4.87035716802883538706812606349, −3.67581315931513537494532436974, −2.70947110218877003576729846505, −1.81725476195762217407518451970, −0.834368080654622395164969259254, 0.834368080654622395164969259254, 1.81725476195762217407518451970, 2.70947110218877003576729846505, 3.67581315931513537494532436974, 4.87035716802883538706812606349, 5.26105147032306915020766840440, 6.03793394555653502998608530406, 6.46393892779863554127241669718, 7.55007030563806169005130121214, 8.125729616692856629127428231228

Graph of the ZZ-function along the critical line