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Properties

Label 2-7400-1.1-c1-0-89
Degree 22
Conductor 74007400
Sign 11
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 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3.27·3-s + 0.447·7-s + 7.69·9-s − 2.50·11-s − 3.23·13-s + 6.79·17-s − 6.54·19-s + 1.46·21-s + 7.94·23-s + 15.3·27-s + 6.66·29-s + 3.96·31-s − 8.18·33-s + 37-s − 10.5·39-s + 5.66·41-s − 5.28·43-s + 7.75·47-s − 6.79·49-s + 22.2·51-s + 4.46·53-s − 21.4·57-s + 9.32·59-s − 12.2·61-s + 3.44·63-s + 1.82·67-s + 25.9·69-s + ⋯
L(s)  = 1  + 1.88·3-s + 0.169·7-s + 2.56·9-s − 0.754·11-s − 0.896·13-s + 1.64·17-s − 1.50·19-s + 0.319·21-s + 1.65·23-s + 2.95·27-s + 1.23·29-s + 0.711·31-s − 1.42·33-s + 0.164·37-s − 1.69·39-s + 0.884·41-s − 0.805·43-s + 1.13·47-s − 0.971·49-s + 3.11·51-s + 0.613·53-s − 2.83·57-s + 1.21·59-s − 1.57·61-s + 0.434·63-s + 0.222·67-s + 3.12·69-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: 11
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: 00
Selberg data: (2, 7400, ( :1/2), 1)(2,\ 7400,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.5877401894.587740189
L(12)L(\frac12) ≈\approx 4.5877401894.587740189
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 1−3.27T+3T2 1 - 3.27T + 3T^{2}
7 1−0.447T+7T2 1 - 0.447T + 7T^{2}
11 1+2.50T+11T2 1 + 2.50T + 11T^{2}
13 1+3.23T+13T2 1 + 3.23T + 13T^{2}
17 1−6.79T+17T2 1 - 6.79T + 17T^{2}
19 1+6.54T+19T2 1 + 6.54T + 19T^{2}
23 1−7.94T+23T2 1 - 7.94T + 23T^{2}
29 1−6.66T+29T2 1 - 6.66T + 29T^{2}
31 1−3.96T+31T2 1 - 3.96T + 31T^{2}
41 1−5.66T+41T2 1 - 5.66T + 41T^{2}
43 1+5.28T+43T2 1 + 5.28T + 43T^{2}
47 1−7.75T+47T2 1 - 7.75T + 47T^{2}
53 1−4.46T+53T2 1 - 4.46T + 53T^{2}
59 1−9.32T+59T2 1 - 9.32T + 59T^{2}
61 1+12.2T+61T2 1 + 12.2T + 61T^{2}
67 1−1.82T+67T2 1 - 1.82T + 67T^{2}
71 1+4.54T+71T2 1 + 4.54T + 71T^{2}
73 1+13.6T+73T2 1 + 13.6T + 73T^{2}
79 1+1.04T+79T2 1 + 1.04T + 79T^{2}
83 1+8.59T+83T2 1 + 8.59T + 83T^{2}
89 1+12.3T+89T2 1 + 12.3T + 89T^{2}
97 1−3.34T+97T2 1 - 3.34T + 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

−8.044506079235692527048631655202, −7.35041764518115123248936761450, −6.91656389008049595352959990643, −5.77944304712094876676754490218, −4.73956642852383867110524656694, −4.34232655924232755951551152009, −3.15944039825729413298451347197, −2.86756513196427945019633809359, −2.06967643671791084909932689054, −1.02409608096304957163464513922, 1.02409608096304957163464513922, 2.06967643671791084909932689054, 2.86756513196427945019633809359, 3.15944039825729413298451347197, 4.34232655924232755951551152009, 4.73956642852383867110524656694, 5.77944304712094876676754490218, 6.91656389008049595352959990643, 7.35041764518115123248936761450, 8.044506079235692527048631655202

Graph of the ZZ-function along the critical line