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Properties

Label 2-7400-1.1-c1-0-70
Degree 22
Conductor 74007400
Sign −1-1
Analytic cond. 59.089259.0892
Root an. cond. 7.686957.68695
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  − 2.95·3-s − 0.939·7-s + 5.74·9-s − 1.84·11-s − 6.00·13-s − 5.08·17-s + 3.73·19-s + 2.77·21-s + 4.23·23-s − 8.13·27-s + 0.310·29-s + 5.50·31-s + 5.45·33-s + 37-s + 17.7·39-s + 5.16·41-s + 8.26·43-s − 11.1·47-s − 6.11·49-s + 15.0·51-s + 12.5·53-s − 11.0·57-s − 10.1·59-s − 12.1·61-s − 5.40·63-s + 8.40·67-s − 12.5·69-s + ⋯
L(s)  = 1  − 1.70·3-s − 0.355·7-s + 1.91·9-s − 0.555·11-s − 1.66·13-s − 1.23·17-s + 0.856·19-s + 0.606·21-s + 0.882·23-s − 1.56·27-s + 0.0576·29-s + 0.988·31-s + 0.948·33-s + 0.164·37-s + 2.84·39-s + 0.806·41-s + 1.26·43-s − 1.63·47-s − 0.873·49-s + 2.10·51-s + 1.72·53-s − 1.46·57-s − 1.32·59-s − 1.55·61-s − 0.680·63-s + 1.02·67-s − 1.50·69-s + ⋯

Functional equation

Λ(s)=(7400s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7400s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 74007400    =    23⋅52⋅372^{3} \cdot 5^{2} \cdot 37
Sign: −1-1
Analytic conductor: 59.089259.0892
Root analytic conductor: 7.686957.68695
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7400, ( :1/2), −1)(2,\ 7400,\ (\ :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 1
37 1−T 1 - T
good3 1+2.95T+3T2 1 + 2.95T + 3T^{2}
7 1+0.939T+7T2 1 + 0.939T + 7T^{2}
11 1+1.84T+11T2 1 + 1.84T + 11T^{2}
13 1+6.00T+13T2 1 + 6.00T + 13T^{2}
17 1+5.08T+17T2 1 + 5.08T + 17T^{2}
19 1−3.73T+19T2 1 - 3.73T + 19T^{2}
23 1−4.23T+23T2 1 - 4.23T + 23T^{2}
29 1−0.310T+29T2 1 - 0.310T + 29T^{2}
31 1−5.50T+31T2 1 - 5.50T + 31T^{2}
41 1−5.16T+41T2 1 - 5.16T + 41T^{2}
43 1−8.26T+43T2 1 - 8.26T + 43T^{2}
47 1+11.1T+47T2 1 + 11.1T + 47T^{2}
53 1−12.5T+53T2 1 - 12.5T + 53T^{2}
59 1+10.1T+59T2 1 + 10.1T + 59T^{2}
61 1+12.1T+61T2 1 + 12.1T + 61T^{2}
67 1−8.40T+67T2 1 - 8.40T + 67T^{2}
71 1−13.3T+71T2 1 - 13.3T + 71T^{2}
73 1+5.06T+73T2 1 + 5.06T + 73T^{2}
79 1−10.3T+79T2 1 - 10.3T + 79T^{2}
83 1+6.69T+83T2 1 + 6.69T + 83T^{2}
89 1−11.2T+89T2 1 - 11.2T + 89T^{2}
97 1−2.46T+97T2 1 - 2.46T + 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

−7.32130639204632746425349171854, −6.75876325076643497393531436269, −6.17960602560708942219713889052, −5.36406284341542676887082043714, −4.83910852924631431464483752500, −4.38621686587478098031226099260, −3.06323099618941609541453997654, −2.20968707240768147442259439429, −0.879741498050797199283884864291, 0, 0.879741498050797199283884864291, 2.20968707240768147442259439429, 3.06323099618941609541453997654, 4.38621686587478098031226099260, 4.83910852924631431464483752500, 5.36406284341542676887082043714, 6.17960602560708942219713889052, 6.75876325076643497393531436269, 7.32130639204632746425349171854

Graph of the ZZ-function along the critical line