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Properties

Label 2-4140-1.1-c1-0-36
Degree 22
Conductor 41404140
Sign −1-1
Analytic cond. 33.058033.0580
Root an. cond. 5.749615.74961
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  + 5-s + 1.44·7-s − 2.44·11-s − 0.449·13-s − 0.550·17-s − 6.44·19-s − 23-s + 25-s + 7.89·29-s − 7·31-s + 1.44·35-s − 8.34·37-s + 1.89·41-s + 0.898·43-s − 2.44·47-s − 4.89·49-s − 10.3·53-s − 2.44·55-s − 7.89·59-s + 9.34·61-s − 0.449·65-s − 3.44·67-s − 3·71-s − 5.34·73-s − 3.55·77-s − 4·79-s − 4.34·83-s + ⋯
L(s)  = 1  + 0.447·5-s + 0.547·7-s − 0.738·11-s − 0.124·13-s − 0.133·17-s − 1.47·19-s − 0.208·23-s + 0.200·25-s + 1.46·29-s − 1.25·31-s + 0.245·35-s − 1.37·37-s + 0.296·41-s + 0.137·43-s − 0.357·47-s − 0.699·49-s − 1.42·53-s − 0.330·55-s − 1.02·59-s + 1.19·61-s − 0.0557·65-s − 0.421·67-s − 0.356·71-s − 0.625·73-s − 0.404·77-s − 0.450·79-s − 0.477·83-s + ⋯

Functional equation

Λ(s)=(4140s/2 ΓC(s) L(s)=(− Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}
Λ(s)=(4140s/2 ΓC(s+1/2) L(s)=(− Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 4140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 41404140    =    22⋅32⋅5⋅232^{2} \cdot 3^{2} \cdot 5 \cdot 23
Sign: −1-1
Analytic conductor: 33.058033.0580
Root analytic conductor: 5.749615.74961
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 11
Selberg data: (2, 4140, ( :1/2), −1)(2,\ 4140,\ (\ :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
3 1 1
5 1−T 1 - T
23 1+T 1 + T
good7 1−1.44T+7T2 1 - 1.44T + 7T^{2}
11 1+2.44T+11T2 1 + 2.44T + 11T^{2}
13 1+0.449T+13T2 1 + 0.449T + 13T^{2}
17 1+0.550T+17T2 1 + 0.550T + 17T^{2}
19 1+6.44T+19T2 1 + 6.44T + 19T^{2}
29 1−7.89T+29T2 1 - 7.89T + 29T^{2}
31 1+7T+31T2 1 + 7T + 31T^{2}
37 1+8.34T+37T2 1 + 8.34T + 37T^{2}
41 1−1.89T+41T2 1 - 1.89T + 41T^{2}
43 1−0.898T+43T2 1 - 0.898T + 43T^{2}
47 1+2.44T+47T2 1 + 2.44T + 47T^{2}
53 1+10.3T+53T2 1 + 10.3T + 53T^{2}
59 1+7.89T+59T2 1 + 7.89T + 59T^{2}
61 1−9.34T+61T2 1 - 9.34T + 61T^{2}
67 1+3.44T+67T2 1 + 3.44T + 67T^{2}
71 1+3T+71T2 1 + 3T + 71T^{2}
73 1+5.34T+73T2 1 + 5.34T + 73T^{2}
79 1+4T+79T2 1 + 4T + 79T^{2}
83 1+4.34T+83T2 1 + 4.34T + 83T^{2}
89 1−7.10T+89T2 1 - 7.10T + 89T^{2}
97 1+5.10T+97T2 1 + 5.10T + 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

−8.184144373530791776156606761460, −7.33403343027783695082205517342, −6.55282847677968303761364243120, −5.84778219585465545404250680930, −4.98642706046647012023421693489, −4.44427505255763618594111199456, −3.31944114817015098863219905235, −2.34908079741243186956872930816, −1.58364564112324276995706147920, 0, 1.58364564112324276995706147920, 2.34908079741243186956872930816, 3.31944114817015098863219905235, 4.44427505255763618594111199456, 4.98642706046647012023421693489, 5.84778219585465545404250680930, 6.55282847677968303761364243120, 7.33403343027783695082205517342, 8.184144373530791776156606761460

Graph of the ZZ-function along the critical line