Properties

Label 2-294-1.1-c3-0-5
Degree 22
Conductor 294294
Sign 11
Analytic cond. 17.346517.3465
Root an. cond. 4.164924.16492
Motivic weight 33
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s − 3·3-s + 4·4-s − 2·5-s − 6·6-s + 8·8-s + 9·9-s − 4·10-s − 8·11-s − 12·12-s + 42·13-s + 6·15-s + 16·16-s + 2·17-s + 18·18-s + 124·19-s − 8·20-s − 16·22-s + 76·23-s − 24·24-s − 121·25-s + 84·26-s − 27·27-s + 254·29-s + 12·30-s + 72·31-s + 32·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 1/2·4-s − 0.178·5-s − 0.408·6-s + 0.353·8-s + 1/3·9-s − 0.126·10-s − 0.219·11-s − 0.288·12-s + 0.896·13-s + 0.103·15-s + 1/4·16-s + 0.0285·17-s + 0.235·18-s + 1.49·19-s − 0.0894·20-s − 0.155·22-s + 0.689·23-s − 0.204·24-s − 0.967·25-s + 0.633·26-s − 0.192·27-s + 1.62·29-s + 0.0730·30-s + 0.417·31-s + 0.176·32-s + ⋯

Functional equation

Λ(s)=(294s/2ΓC(s)L(s)=(Λ(4s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}
Λ(s)=(294s/2ΓC(s+3/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 294 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 294294    =    23722 \cdot 3 \cdot 7^{2}
Sign: 11
Analytic conductor: 17.346517.3465
Root analytic conductor: 4.164924.16492
Motivic weight: 33
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 294, ( :3/2), 1)(2,\ 294,\ (\ :3/2),\ 1)

Particular Values

L(2)L(2) \approx 2.4448774032.444877403
L(12)L(\frac12) \approx 2.4448774032.444877403
L(52)L(\frac{5}{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 1pT 1 - p T
3 1+pT 1 + p T
7 1 1
good5 1+2T+p3T2 1 + 2 T + p^{3} T^{2}
11 1+8T+p3T2 1 + 8 T + p^{3} T^{2}
13 142T+p3T2 1 - 42 T + p^{3} T^{2}
17 12T+p3T2 1 - 2 T + p^{3} T^{2}
19 1124T+p3T2 1 - 124 T + p^{3} T^{2}
23 176T+p3T2 1 - 76 T + p^{3} T^{2}
29 1254T+p3T2 1 - 254 T + p^{3} T^{2}
31 172T+p3T2 1 - 72 T + p^{3} T^{2}
37 1398T+p3T2 1 - 398 T + p^{3} T^{2}
41 1+462T+p3T2 1 + 462 T + p^{3} T^{2}
43 1212T+p3T2 1 - 212 T + p^{3} T^{2}
47 1264T+p3T2 1 - 264 T + p^{3} T^{2}
53 1+162T+p3T2 1 + 162 T + p^{3} T^{2}
59 1772T+p3T2 1 - 772 T + p^{3} T^{2}
61 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
67 1+764T+p3T2 1 + 764 T + p^{3} T^{2}
71 1+236T+p3T2 1 + 236 T + p^{3} T^{2}
73 1+418T+p3T2 1 + 418 T + p^{3} T^{2}
79 1552T+p3T2 1 - 552 T + p^{3} T^{2}
83 1+1036T+p3T2 1 + 1036 T + p^{3} T^{2}
89 1+30T+p3T2 1 + 30 T + p^{3} T^{2}
97 11190T+p3T2 1 - 1190 T + p^{3} T^{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

−11.56886698311922542653532608853, −10.63851879848623807711697193557, −9.673140973884217385418283947736, −8.299091446871014868611931534960, −7.24487164749901783641199687735, −6.20217852812527652318977227994, −5.30256143482064034171982614073, −4.20281487194666214110630342027, −2.94703327487831006010805444320, −1.09334361875456715075271536774, 1.09334361875456715075271536774, 2.94703327487831006010805444320, 4.20281487194666214110630342027, 5.30256143482064034171982614073, 6.20217852812527652318977227994, 7.24487164749901783641199687735, 8.299091446871014868611931534960, 9.673140973884217385418283947736, 10.63851879848623807711697193557, 11.56886698311922542653532608853

Graph of the ZZ-function along the critical line