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Properties

Label 2-7600-1.1-c1-0-80
Degree 22
Conductor 76007600
Sign −1-1
Analytic cond. 60.686360.6863
Root an. cond. 7.790147.79014
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  − 2.25·3-s − 4.22·7-s + 2.08·9-s + 5.13·11-s + 3.16·13-s − 6.48·17-s + 19-s + 9.53·21-s − 7.56·23-s + 2.05·27-s + 0.832·29-s + 4.51·31-s − 11.5·33-s + 0.137·37-s − 7.14·39-s − 11.6·41-s + 2.51·43-s + 5.96·47-s + 10.8·49-s + 14.6·51-s + 0.225·53-s − 2.25·57-s − 5.39·59-s + 14.4·61-s − 8.82·63-s + 4.11·67-s + 17.0·69-s + ⋯
L(s)  = 1  − 1.30·3-s − 1.59·7-s + 0.695·9-s + 1.54·11-s + 0.878·13-s − 1.57·17-s + 0.229·19-s + 2.07·21-s − 1.57·23-s + 0.395·27-s + 0.154·29-s + 0.810·31-s − 2.01·33-s + 0.0226·37-s − 1.14·39-s − 1.81·41-s + 0.382·43-s + 0.869·47-s + 1.55·49-s + 2.04·51-s + 0.0309·53-s − 0.298·57-s − 0.702·59-s + 1.85·61-s − 1.11·63-s + 0.502·67-s + 2.05·69-s + ⋯

Functional equation

Λ(s)=(7600s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(7600s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 7600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 76007600    =    24⋅52⋅192^{4} \cdot 5^{2} \cdot 19
Sign: −1-1
Analytic conductor: 60.686360.6863
Root analytic conductor: 7.790147.79014
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 7600, ( :1/2), −1)(2,\ 7600,\ (\ :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
19 1−T 1 - T
good3 1+2.25T+3T2 1 + 2.25T + 3T^{2}
7 1+4.22T+7T2 1 + 4.22T + 7T^{2}
11 1−5.13T+11T2 1 - 5.13T + 11T^{2}
13 1−3.16T+13T2 1 - 3.16T + 13T^{2}
17 1+6.48T+17T2 1 + 6.48T + 17T^{2}
23 1+7.56T+23T2 1 + 7.56T + 23T^{2}
29 1−0.832T+29T2 1 - 0.832T + 29T^{2}
31 1−4.51T+31T2 1 - 4.51T + 31T^{2}
37 1−0.137T+37T2 1 - 0.137T + 37T^{2}
41 1+11.6T+41T2 1 + 11.6T + 41T^{2}
43 1−2.51T+43T2 1 - 2.51T + 43T^{2}
47 1−5.96T+47T2 1 - 5.96T + 47T^{2}
53 1−0.225T+53T2 1 - 0.225T + 53T^{2}
59 1+5.39T+59T2 1 + 5.39T + 59T^{2}
61 1−14.4T+61T2 1 - 14.4T + 61T^{2}
67 1−4.11T+67T2 1 - 4.11T + 67T^{2}
71 1+3.82T+71T2 1 + 3.82T + 71T^{2}
73 1−4.70T+73T2 1 - 4.70T + 73T^{2}
79 1+10.6T+79T2 1 + 10.6T + 79T^{2}
83 1−12.0T+83T2 1 - 12.0T + 83T^{2}
89 1+10T+89T2 1 + 10T + 89T^{2}
97 1−3.93T+97T2 1 - 3.93T + 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

−7.03713252093506338018453144333, −6.65813023076472789924135934952, −6.16592436759387901156598238660, −5.80032014915015338544985018101, −4.64678711959901529819808212148, −3.97831497810698627751373789667, −3.35573897044826210629920924165, −2.13990374872545626816333936382, −0.949037716635654469383738311356, 0, 0.949037716635654469383738311356, 2.13990374872545626816333936382, 3.35573897044826210629920924165, 3.97831497810698627751373789667, 4.64678711959901529819808212148, 5.80032014915015338544985018101, 6.16592436759387901156598238660, 6.65813023076472789924135934952, 7.03713252093506338018453144333

Graph of the ZZ-function along the critical line