Properties

Label 2-6336-1.1-c1-0-0
Degree 22
Conductor 63366336
Sign 11
Analytic cond. 50.593250.5932
Root an. cond. 7.112897.11289
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  − 0.561·5-s − 5.12·7-s − 11-s − 3.12·13-s − 2·17-s + 4·19-s − 6.56·23-s − 4.68·25-s + 3.12·29-s − 1.43·31-s + 2.87·35-s + 3.43·37-s − 7.12·41-s − 1.12·43-s − 8·47-s + 19.2·49-s − 4.24·53-s + 0.561·55-s − 12.8·59-s + 7.12·61-s + 1.75·65-s − 5.43·67-s − 3.68·71-s − 3.12·73-s + 5.12·77-s − 2.87·79-s + 9.12·83-s + ⋯
L(s)  = 1  − 0.251·5-s − 1.93·7-s − 0.301·11-s − 0.866·13-s − 0.485·17-s + 0.917·19-s − 1.36·23-s − 0.936·25-s + 0.579·29-s − 0.258·31-s + 0.486·35-s + 0.565·37-s − 1.11·41-s − 0.171·43-s − 1.16·47-s + 2.74·49-s − 0.583·53-s + 0.0757·55-s − 1.66·59-s + 0.912·61-s + 0.217·65-s − 0.664·67-s − 0.437·71-s − 0.365·73-s + 0.583·77-s − 0.323·79-s + 1.00·83-s + ⋯

Functional equation

Λ(s)=(6336s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(6336s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 6336 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 63366336    =    2632112^{6} \cdot 3^{2} \cdot 11
Sign: 11
Analytic conductor: 50.593250.5932
Root analytic conductor: 7.112897.11289
Motivic weight: 11
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 6336, ( :1/2), 1)(2,\ 6336,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 0.51593384350.5159338435
L(12)L(\frac12) \approx 0.51593384350.5159338435
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=pFp(ps)1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppFp(T)F_p(T)
bad2 1 1
3 1 1
11 1+T 1 + T
good5 1+0.561T+5T2 1 + 0.561T + 5T^{2}
7 1+5.12T+7T2 1 + 5.12T + 7T^{2}
13 1+3.12T+13T2 1 + 3.12T + 13T^{2}
17 1+2T+17T2 1 + 2T + 17T^{2}
19 14T+19T2 1 - 4T + 19T^{2}
23 1+6.56T+23T2 1 + 6.56T + 23T^{2}
29 13.12T+29T2 1 - 3.12T + 29T^{2}
31 1+1.43T+31T2 1 + 1.43T + 31T^{2}
37 13.43T+37T2 1 - 3.43T + 37T^{2}
41 1+7.12T+41T2 1 + 7.12T + 41T^{2}
43 1+1.12T+43T2 1 + 1.12T + 43T^{2}
47 1+8T+47T2 1 + 8T + 47T^{2}
53 1+4.24T+53T2 1 + 4.24T + 53T^{2}
59 1+12.8T+59T2 1 + 12.8T + 59T^{2}
61 17.12T+61T2 1 - 7.12T + 61T^{2}
67 1+5.43T+67T2 1 + 5.43T + 67T^{2}
71 1+3.68T+71T2 1 + 3.68T + 71T^{2}
73 1+3.12T+73T2 1 + 3.12T + 73T^{2}
79 1+2.87T+79T2 1 + 2.87T + 79T^{2}
83 19.12T+83T2 1 - 9.12T + 83T^{2}
89 19.68T+89T2 1 - 9.68T + 89T^{2}
97 111.4T+97T2 1 - 11.4T + 97T^{2}
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   L(s)=p j=12(1αj,pps)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

−7.84960030077535263021306549704, −7.38182180226491932793983993091, −6.49375406684251816548198980593, −6.12615412230479030103308315268, −5.22049160386946861150690571969, −4.32733388686746680024234025036, −3.46975669874966716870517664579, −2.93720590940063199864601979211, −1.97344929397874887580778724500, −0.34799344596824056453445742882, 0.34799344596824056453445742882, 1.97344929397874887580778724500, 2.93720590940063199864601979211, 3.46975669874966716870517664579, 4.32733388686746680024234025036, 5.22049160386946861150690571969, 6.12615412230479030103308315268, 6.49375406684251816548198980593, 7.38182180226491932793983993091, 7.84960030077535263021306549704

Graph of the ZZ-function along the critical line