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Properties

Label 2-1450-1.1-c1-0-30
Degree 22
Conductor 14501450
Sign −1-1
Analytic cond. 11.578311.5783
Root an. cond. 3.402693.40269
Motivic weight 11
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank 11

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 2.30·3-s + 4-s − 2.30·6-s − 0.697·7-s + 8-s + 2.30·9-s + 2.60·11-s − 2.30·12-s − 6.30·13-s − 0.697·14-s + 16-s + 3.90·17-s + 2.30·18-s − 0.605·19-s + 1.60·21-s + 2.60·22-s − 1.69·23-s − 2.30·24-s − 6.30·26-s + 1.60·27-s − 0.697·28-s + 29-s − 7.90·31-s + 32-s − 6·33-s + 3.90·34-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.32·3-s + 0.5·4-s − 0.940·6-s − 0.263·7-s + 0.353·8-s + 0.767·9-s + 0.785·11-s − 0.664·12-s − 1.74·13-s − 0.186·14-s + 0.250·16-s + 0.947·17-s + 0.542·18-s − 0.138·19-s + 0.350·21-s + 0.555·22-s − 0.353·23-s − 0.470·24-s − 1.23·26-s + 0.308·27-s − 0.131·28-s + 0.185·29-s − 1.42·31-s + 0.176·32-s − 1.04·33-s + 0.670·34-s + ⋯

Functional equation

Λ(s)=(1450s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1450s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 14501450    =    2⋅52⋅292 \cdot 5^{2} \cdot 29
Sign: −1-1
Analytic conductor: 11.578311.5783
Root analytic conductor: 3.402693.40269
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1450, ( :1/2), −1)(2,\ 1450,\ (\ :1/2),\ -1)

Particular Values

L(1)L(1) == 00
L(12)L(\frac12) == 00
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−T 1 - T
5 1 1
29 1−T 1 - T
good3 1+2.30T+3T2 1 + 2.30T + 3T^{2}
7 1+0.697T+7T2 1 + 0.697T + 7T^{2}
11 1−2.60T+11T2 1 - 2.60T + 11T^{2}
13 1+6.30T+13T2 1 + 6.30T + 13T^{2}
17 1−3.90T+17T2 1 - 3.90T + 17T^{2}
19 1+0.605T+19T2 1 + 0.605T + 19T^{2}
23 1+1.69T+23T2 1 + 1.69T + 23T^{2}
31 1+7.90T+31T2 1 + 7.90T + 31T^{2}
37 1+9.81T+37T2 1 + 9.81T + 37T^{2}
41 1+8.60T+41T2 1 + 8.60T + 41T^{2}
43 1+3.30T+43T2 1 + 3.30T + 43T^{2}
47 1+47T2 1 + 47T^{2}
53 1−3.90T+53T2 1 - 3.90T + 53T^{2}
59 1+0.908T+59T2 1 + 0.908T + 59T^{2}
61 1+3.09T+61T2 1 + 3.09T + 61T^{2}
67 1−9.21T+67T2 1 - 9.21T + 67T^{2}
71 1+71T2 1 + 71T^{2}
73 1+2.51T+73T2 1 + 2.51T + 73T^{2}
79 1+4.90T+79T2 1 + 4.90T + 79T^{2}
83 1−8.60T+83T2 1 - 8.60T + 83T^{2}
89 1+14.6T+89T2 1 + 14.6T + 89T^{2}
97 1+1.09T+97T2 1 + 1.09T + 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

−9.407990081614753518438997617253, −8.122977075012076541251019641000, −6.99003145307253373700181367280, −6.70473469805233267182206547619, −5.49718281366288437666415243477, −5.22680105345781307133436841878, −4.18060447012509745400533690702, −3.12032659178808281312473073020, −1.69465286739611247746851171454, 0, 1.69465286739611247746851171454, 3.12032659178808281312473073020, 4.18060447012509745400533690702, 5.22680105345781307133436841878, 5.49718281366288437666415243477, 6.70473469805233267182206547619, 6.99003145307253373700181367280, 8.122977075012076541251019641000, 9.407990081614753518438997617253

Graph of the ZZ-function along the critical line