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Properties

Label 2-7448-1.1-c1-0-75
Degree 22
Conductor 74487448
Sign −1-1
Analytic cond. 59.472559.4725
Root an. cond. 7.711847.71184
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  − 2.30·3-s − 2.57·5-s + 2.30·9-s − 1.35·11-s + 4.19·13-s + 5.92·15-s − 4.92·17-s − 19-s + 6.19·23-s + 1.62·25-s + 1.60·27-s + 0.760·29-s − 9.46·31-s + 3.11·33-s − 5.27·37-s − 9.66·39-s − 1.23·41-s − 9.91·43-s − 5.92·45-s + 5.77·47-s + 11.3·51-s − 1.15·53-s + 3.48·55-s + 2.30·57-s + 7.88·59-s + 8.31·61-s − 10.8·65-s + ⋯
L(s)  = 1  − 1.32·3-s − 1.15·5-s + 0.767·9-s − 0.407·11-s + 1.16·13-s + 1.53·15-s − 1.19·17-s − 0.229·19-s + 1.29·23-s + 0.324·25-s + 0.308·27-s + 0.141·29-s − 1.70·31-s + 0.542·33-s − 0.867·37-s − 1.54·39-s − 0.193·41-s − 1.51·43-s − 0.883·45-s + 0.841·47-s + 1.58·51-s − 0.158·53-s + 0.469·55-s + 0.305·57-s + 1.02·59-s + 1.06·61-s − 1.33·65-s + ⋯

Functional equation

Λ(s)=(7448s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7448s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74487448    =    23⋅72⋅192^{3} \cdot 7^{2} \cdot 19
Sign: −1-1
Analytic conductor: 59.472559.4725
Root analytic conductor: 7.711847.71184
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7448, ( :1/2), −1)(2,\ 7448,\ (\ :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 1
19 1+T 1 + T
good3 1+2.30T+3T2 1 + 2.30T + 3T^{2}
5 1+2.57T+5T2 1 + 2.57T + 5T^{2}
11 1+1.35T+11T2 1 + 1.35T + 11T^{2}
13 1−4.19T+13T2 1 - 4.19T + 13T^{2}
17 1+4.92T+17T2 1 + 4.92T + 17T^{2}
23 1−6.19T+23T2 1 - 6.19T + 23T^{2}
29 1−0.760T+29T2 1 - 0.760T + 29T^{2}
31 1+9.46T+31T2 1 + 9.46T + 31T^{2}
37 1+5.27T+37T2 1 + 5.27T + 37T^{2}
41 1+1.23T+41T2 1 + 1.23T + 41T^{2}
43 1+9.91T+43T2 1 + 9.91T + 43T^{2}
47 1−5.77T+47T2 1 - 5.77T + 47T^{2}
53 1+1.15T+53T2 1 + 1.15T + 53T^{2}
59 1−7.88T+59T2 1 - 7.88T + 59T^{2}
61 1−8.31T+61T2 1 - 8.31T + 61T^{2}
67 1−15.3T+67T2 1 - 15.3T + 67T^{2}
71 1−6.56T+71T2 1 - 6.56T + 71T^{2}
73 1−9.02T+73T2 1 - 9.02T + 73T^{2}
79 1+0.707T+79T2 1 + 0.707T + 79T^{2}
83 1−11.6T+83T2 1 - 11.6T + 83T^{2}
89 1+4.72T+89T2 1 + 4.72T + 89T^{2}
97 1−12.4T+97T2 1 - 12.4T + 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

−7.29967634268351991800769930147, −6.83226236620644281526252641048, −6.22622474749822999103165984325, −5.27902990897288876679113310043, −4.93988102610281771554963232699, −3.92209407907732809652277424733, −3.46241559926110444637607698225, −2.13574436856237165383439008399, −0.875257730752836817094680420505, 0, 0.875257730752836817094680420505, 2.13574436856237165383439008399, 3.46241559926110444637607698225, 3.92209407907732809652277424733, 4.93988102610281771554963232699, 5.27902990897288876679113310043, 6.22622474749822999103165984325, 6.83226236620644281526252641048, 7.29967634268351991800769930147

Graph of the ZZ-function along the critical line