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Properties

Label 2-1064-1.1-c1-0-23
Degree 22
Conductor 10641064
Sign −1-1
Analytic cond. 8.496088.49608
Root an. cond. 2.914802.91480
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  + 1.30·3-s − 5-s − 7-s − 1.30·9-s − 2.30·11-s − 3.60·13-s − 1.30·15-s + 1.69·17-s + 19-s − 1.30·21-s − 0.394·23-s − 4·25-s − 5.60·27-s − 4.30·29-s − 8.30·31-s − 3·33-s + 35-s + 3.60·37-s − 4.69·39-s + 0.302·41-s + 7.21·43-s + 1.30·45-s − 7.60·47-s + 49-s + 2.21·51-s − 3.90·53-s + 2.30·55-s + ⋯
L(s)  = 1  + 0.752·3-s − 0.447·5-s − 0.377·7-s − 0.434·9-s − 0.694·11-s − 1.00·13-s − 0.336·15-s + 0.411·17-s + 0.229·19-s − 0.284·21-s − 0.0822·23-s − 0.800·25-s − 1.07·27-s − 0.799·29-s − 1.49·31-s − 0.522·33-s + 0.169·35-s + 0.592·37-s − 0.752·39-s + 0.0472·41-s + 1.09·43-s + 0.194·45-s − 1.10·47-s + 0.142·49-s + 0.309·51-s − 0.536·53-s + 0.310·55-s + ⋯

Functional equation

Λ(s)=(1064s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(1064s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 10641064    =    23⋅7⋅192^{3} \cdot 7 \cdot 19
Sign: −1-1
Analytic conductor: 8.496088.49608
Root analytic conductor: 2.914802.91480
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 1064, ( :1/2), −1)(2,\ 1064,\ (\ :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 1
7 1+T 1 + T
19 1−T 1 - T
good3 1−1.30T+3T2 1 - 1.30T + 3T^{2}
5 1+T+5T2 1 + T + 5T^{2}
11 1+2.30T+11T2 1 + 2.30T + 11T^{2}
13 1+3.60T+13T2 1 + 3.60T + 13T^{2}
17 1−1.69T+17T2 1 - 1.69T + 17T^{2}
23 1+0.394T+23T2 1 + 0.394T + 23T^{2}
29 1+4.30T+29T2 1 + 4.30T + 29T^{2}
31 1+8.30T+31T2 1 + 8.30T + 31T^{2}
37 1−3.60T+37T2 1 - 3.60T + 37T^{2}
41 1−0.302T+41T2 1 - 0.302T + 41T^{2}
43 1−7.21T+43T2 1 - 7.21T + 43T^{2}
47 1+7.60T+47T2 1 + 7.60T + 47T^{2}
53 1+3.90T+53T2 1 + 3.90T + 53T^{2}
59 1−5.60T+59T2 1 - 5.60T + 59T^{2}
61 1−8.21T+61T2 1 - 8.21T + 61T^{2}
67 1+10.9T+67T2 1 + 10.9T + 67T^{2}
71 1−8.81T+71T2 1 - 8.81T + 71T^{2}
73 1+5.90T+73T2 1 + 5.90T + 73T^{2}
79 1+14T+79T2 1 + 14T + 79T^{2}
83 1+10.5T+83T2 1 + 10.5T + 83T^{2}
89 1−13.8T+89T2 1 - 13.8T + 89T^{2}
97 1−16.8T+97T2 1 - 16.8T + 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

−9.463460236620376338393224653993, −8.663661194119957934342112649897, −7.67202475509365774400635116925, −7.41163122871988559973614870312, −5.98612541146469300117065519906, −5.18202102813611815477541820396, −3.93945824571502802441167542365, −3.08023459896184746060602119412, −2.13950487143381518025975937095, 0, 2.13950487143381518025975937095, 3.08023459896184746060602119412, 3.93945824571502802441167542365, 5.18202102813611815477541820396, 5.98612541146469300117065519906, 7.41163122871988559973614870312, 7.67202475509365774400635116925, 8.663661194119957934342112649897, 9.463460236620376338393224653993

Graph of the ZZ-function along the critical line