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Properties

Label 2-4304-1.1-c1-0-122
Degree 22
Conductor 43044304
Sign −1-1
Analytic cond. 34.367634.3676
Root an. cond. 5.862385.86238
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  + 1.37·3-s + 1.97·5-s − 2.44·7-s − 1.09·9-s − 3.43·11-s + 4.69·13-s + 2.72·15-s + 1.90·17-s + 1.23·19-s − 3.36·21-s − 6.49·23-s − 1.10·25-s − 5.65·27-s − 5.54·29-s − 4.88·31-s − 4.73·33-s − 4.81·35-s − 10.2·37-s + 6.47·39-s − 5.60·41-s + 2.85·43-s − 2.16·45-s + 1.80·47-s − 1.04·49-s + 2.62·51-s + 2.04·53-s − 6.77·55-s + ⋯
L(s)  = 1  + 0.796·3-s + 0.882·5-s − 0.922·7-s − 0.366·9-s − 1.03·11-s + 1.30·13-s + 0.702·15-s + 0.461·17-s + 0.282·19-s − 0.734·21-s − 1.35·23-s − 0.221·25-s − 1.08·27-s − 1.02·29-s − 0.876·31-s − 0.824·33-s − 0.814·35-s − 1.68·37-s + 1.03·39-s − 0.875·41-s + 0.434·43-s − 0.322·45-s + 0.264·47-s − 0.148·49-s + 0.367·51-s + 0.280·53-s − 0.914·55-s + ⋯

Functional equation

Λ(s)=(4304s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4304s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4304 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 43044304    =    24⋅2692^{4} \cdot 269
Sign: −1-1
Analytic conductor: 34.367634.3676
Root analytic conductor: 5.862385.86238
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4304, ( :1/2), −1)(2,\ 4304,\ (\ :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
269 1−T 1 - T
good3 1−1.37T+3T2 1 - 1.37T + 3T^{2}
5 1−1.97T+5T2 1 - 1.97T + 5T^{2}
7 1+2.44T+7T2 1 + 2.44T + 7T^{2}
11 1+3.43T+11T2 1 + 3.43T + 11T^{2}
13 1−4.69T+13T2 1 - 4.69T + 13T^{2}
17 1−1.90T+17T2 1 - 1.90T + 17T^{2}
19 1−1.23T+19T2 1 - 1.23T + 19T^{2}
23 1+6.49T+23T2 1 + 6.49T + 23T^{2}
29 1+5.54T+29T2 1 + 5.54T + 29T^{2}
31 1+4.88T+31T2 1 + 4.88T + 31T^{2}
37 1+10.2T+37T2 1 + 10.2T + 37T^{2}
41 1+5.60T+41T2 1 + 5.60T + 41T^{2}
43 1−2.85T+43T2 1 - 2.85T + 43T^{2}
47 1−1.80T+47T2 1 - 1.80T + 47T^{2}
53 1−2.04T+53T2 1 - 2.04T + 53T^{2}
59 1−1.02T+59T2 1 - 1.02T + 59T^{2}
61 1−12.6T+61T2 1 - 12.6T + 61T^{2}
67 1+1.65T+67T2 1 + 1.65T + 67T^{2}
71 1−4.59T+71T2 1 - 4.59T + 71T^{2}
73 1−5.17T+73T2 1 - 5.17T + 73T^{2}
79 1+13.5T+79T2 1 + 13.5T + 79T^{2}
83 1−13.9T+83T2 1 - 13.9T + 83T^{2}
89 1+9.64T+89T2 1 + 9.64T + 89T^{2}
97 1+16.2T+97T2 1 + 16.2T + 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

−8.177187871039468361191662247051, −7.38696311022926448605361549916, −6.47800471872512986131415016551, −5.65800348733173796378417175147, −5.44045495767852663358880163890, −3.79704051966393730680536035594, −3.42424418399327095164634090606, −2.44491105675455185585088522138, −1.71091608164255320629422460618, 0, 1.71091608164255320629422460618, 2.44491105675455185585088522138, 3.42424418399327095164634090606, 3.79704051966393730680536035594, 5.44045495767852663358880163890, 5.65800348733173796378417175147, 6.47800471872512986131415016551, 7.38696311022926448605361549916, 8.177187871039468361191662247051

Graph of the ZZ-function along the critical line