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Properties

Label 2-9280-1.1-c1-0-121
Degree 22
Conductor 92809280
Sign −1-1
Analytic cond. 74.101174.1011
Root an. cond. 8.608208.60820
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  − 1.35·3-s + 5-s − 0.648·7-s − 1.17·9-s − 3.35·11-s − 4.17·13-s − 1.35·15-s + 4.82·17-s − 6.82·19-s + 0.876·21-s + 5.52·23-s + 25-s + 5.64·27-s + 29-s − 2.82·31-s + 4.53·33-s − 0.648·35-s + 10.2·37-s + 5.64·39-s + 8.17·41-s + 5.69·43-s − 1.17·45-s − 2.64·47-s − 6.58·49-s − 6.51·51-s + 2.87·53-s − 3.35·55-s + ⋯
L(s)  = 1  − 0.780·3-s + 0.447·5-s − 0.244·7-s − 0.390·9-s − 1.01·11-s − 1.15·13-s − 0.349·15-s + 1.16·17-s − 1.56·19-s + 0.191·21-s + 1.15·23-s + 0.200·25-s + 1.08·27-s + 0.185·29-s − 0.506·31-s + 0.788·33-s − 0.109·35-s + 1.68·37-s + 0.903·39-s + 1.27·41-s + 0.868·43-s − 0.174·45-s − 0.386·47-s − 0.940·49-s − 0.912·51-s + 0.395·53-s − 0.451·55-s + ⋯

Functional equation

Λ(s)=(9280s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(9280s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 9280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 92809280    =    26⋅5⋅292^{6} \cdot 5 \cdot 29
Sign: −1-1
Analytic conductor: 74.101174.1011
Root analytic conductor: 8.608208.60820
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 9280, ( :1/2), −1)(2,\ 9280,\ (\ :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
5 1−T 1 - T
29 1−T 1 - T
good3 1+1.35T+3T2 1 + 1.35T + 3T^{2}
7 1+0.648T+7T2 1 + 0.648T + 7T^{2}
11 1+3.35T+11T2 1 + 3.35T + 11T^{2}
13 1+4.17T+13T2 1 + 4.17T + 13T^{2}
17 1−4.82T+17T2 1 - 4.82T + 17T^{2}
19 1+6.82T+19T2 1 + 6.82T + 19T^{2}
23 1−5.52T+23T2 1 - 5.52T + 23T^{2}
31 1+2.82T+31T2 1 + 2.82T + 31T^{2}
37 1−10.2T+37T2 1 - 10.2T + 37T^{2}
41 1−8.17T+41T2 1 - 8.17T + 41T^{2}
43 1−5.69T+43T2 1 - 5.69T + 43T^{2}
47 1+2.64T+47T2 1 + 2.64T + 47T^{2}
53 1−2.87T+53T2 1 - 2.87T + 53T^{2}
59 1−13.2T+59T2 1 - 13.2T + 59T^{2}
61 1−1.12T+61T2 1 - 1.12T + 61T^{2}
67 1+1.52T+67T2 1 + 1.52T + 67T^{2}
71 1+8.87T+71T2 1 + 8.87T + 71T^{2}
73 1−9.69T+73T2 1 - 9.69T + 73T^{2}
79 1−8.99T+79T2 1 - 8.99T + 79T^{2}
83 1+1.94T+83T2 1 + 1.94T + 83T^{2}
89 1−17.0T+89T2 1 - 17.0T + 89T^{2}
97 1+13.3T+97T2 1 + 13.3T + 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

−7.36474845551844160442939395196, −6.52184939675738630649343147919, −5.98705323828913852190704659362, −5.26531430905038674971687544940, −4.91043433286482236565443491363, −3.92986168420535948073313371013, −2.70424092354768665629015235224, −2.47429003784572484998331690856, −0.999637294841537897974002833620, 0, 0.999637294841537897974002833620, 2.47429003784572484998331690856, 2.70424092354768665629015235224, 3.92986168420535948073313371013, 4.91043433286482236565443491363, 5.26531430905038674971687544940, 5.98705323828913852190704659362, 6.52184939675738630649343147919, 7.36474845551844160442939395196

Graph of the ZZ-function along the critical line