Properties

Label 2-616-1.1-c1-0-6
Degree 22
Conductor 616616
Sign 11
Analytic cond. 4.918784.91878
Root an. cond. 2.217832.21783
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 2·5-s + 7-s + 9-s − 11-s + 4·15-s + 4·17-s + 4·19-s + 2·21-s − 4·23-s − 25-s − 4·27-s + 2·29-s − 2·31-s − 2·33-s + 2·35-s − 6·37-s + 4·41-s − 4·43-s + 2·45-s + 2·47-s + 49-s + 8·51-s + 2·53-s − 2·55-s + 8·57-s − 6·59-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.894·5-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 1.03·15-s + 0.970·17-s + 0.917·19-s + 0.436·21-s − 0.834·23-s − 1/5·25-s − 0.769·27-s + 0.371·29-s − 0.359·31-s − 0.348·33-s + 0.338·35-s − 0.986·37-s + 0.624·41-s − 0.609·43-s + 0.298·45-s + 0.291·47-s + 1/7·49-s + 1.12·51-s + 0.274·53-s − 0.269·55-s + 1.05·57-s − 0.781·59-s + ⋯

Functional equation

Λ(s)=(616s/2ΓC(s)L(s)=(Λ(2s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(616s/2ΓC(s+1/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 616 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 616616    =    237112^{3} \cdot 7 \cdot 11
Sign: 11
Analytic conductor: 4.918784.91878
Root analytic conductor: 2.217832.21783
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 616, ( :1/2), 1)(2,\ 616,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) \approx 2.4835006732.483500673
L(12)L(\frac12) \approx 2.4835006732.483500673
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)Isogeny Class over Fp\mathbf{F}_p
bad2 1 1
7 1T 1 - T
11 1+T 1 + T
good3 12T+pT2 1 - 2 T + p T^{2} 1.3.ac
5 12T+pT2 1 - 2 T + p T^{2} 1.5.ac
13 1+pT2 1 + p T^{2} 1.13.a
17 14T+pT2 1 - 4 T + p T^{2} 1.17.ae
19 14T+pT2 1 - 4 T + p T^{2} 1.19.ae
23 1+4T+pT2 1 + 4 T + p T^{2} 1.23.e
29 12T+pT2 1 - 2 T + p T^{2} 1.29.ac
31 1+2T+pT2 1 + 2 T + p T^{2} 1.31.c
37 1+6T+pT2 1 + 6 T + p T^{2} 1.37.g
41 14T+pT2 1 - 4 T + p T^{2} 1.41.ae
43 1+4T+pT2 1 + 4 T + p T^{2} 1.43.e
47 12T+pT2 1 - 2 T + p T^{2} 1.47.ac
53 12T+pT2 1 - 2 T + p T^{2} 1.53.ac
59 1+6T+pT2 1 + 6 T + p T^{2} 1.59.g
61 14T+pT2 1 - 4 T + p T^{2} 1.61.ae
67 1+pT2 1 + p T^{2} 1.67.a
71 1+12T+pT2 1 + 12 T + p T^{2} 1.71.m
73 116T+pT2 1 - 16 T + p T^{2} 1.73.aq
79 1+8T+pT2 1 + 8 T + p T^{2} 1.79.i
83 1+12T+pT2 1 + 12 T + p T^{2} 1.83.m
89 110T+pT2 1 - 10 T + p T^{2} 1.89.ak
97 1+2T+pT2 1 + 2 T + p T^{2} 1.97.c
show more
show less
   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

−10.28981135359909303761462495429, −9.721026942562876571917863787048, −8.900267729944967573194117141070, −8.051352337308495880478764667111, −7.35767416332621097285914793198, −5.98413065916960487951751915735, −5.19667268136125807114507816372, −3.75056949088258166882410190362, −2.72800026016992914882879677292, −1.66644659770392612638218462013, 1.66644659770392612638218462013, 2.72800026016992914882879677292, 3.75056949088258166882410190362, 5.19667268136125807114507816372, 5.98413065916960487951751915735, 7.35767416332621097285914793198, 8.051352337308495880478764667111, 8.900267729944967573194117141070, 9.721026942562876571917863787048, 10.28981135359909303761462495429

Graph of the ZZ-function along the critical line