Properties

Label 2-560-1.1-c7-0-50
Degree 22
Conductor 560560
Sign 11
Analytic cond. 174.935174.935
Root an. cond. 13.226313.2263
Motivic weight 77
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 93·3-s + 125·5-s − 343·7-s + 6.46e3·9-s + 2.16e3·11-s − 1.66e3·13-s + 1.16e4·15-s − 3.57e4·17-s − 2.02e4·19-s − 3.18e4·21-s + 4.21e4·23-s + 1.56e4·25-s + 3.97e5·27-s − 1.11e5·29-s + 2.69e5·31-s + 2.01e5·33-s − 4.28e4·35-s + 5.32e5·37-s − 1.54e5·39-s + 1.58e5·41-s + 5.21e5·43-s + 8.07e5·45-s + 9.39e5·47-s + 1.17e5·49-s − 3.32e6·51-s − 4.08e5·53-s + 2.70e5·55-s + ⋯
L(s)  = 1  + 1.98·3-s + 0.447·5-s − 0.377·7-s + 2.95·9-s + 0.490·11-s − 0.209·13-s + 0.889·15-s − 1.76·17-s − 0.676·19-s − 0.751·21-s + 0.722·23-s + 1/5·25-s + 3.88·27-s − 0.851·29-s + 1.62·31-s + 0.976·33-s − 0.169·35-s + 1.72·37-s − 0.416·39-s + 0.358·41-s + 1.00·43-s + 1.32·45-s + 1.32·47-s + 1/7·49-s − 3.51·51-s − 0.376·53-s + 0.219·55-s + ⋯

Functional equation

Λ(s)=(560s/2ΓC(s)L(s)=(Λ(8s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}
Λ(s)=(560s/2ΓC(s+7/2)L(s)=(Λ(1s)\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 560560    =    24572^{4} \cdot 5 \cdot 7
Sign: 11
Analytic conductor: 174.935174.935
Root analytic conductor: 13.226313.2263
Motivic weight: 77
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 560, ( :7/2), 1)(2,\ 560,\ (\ :7/2),\ 1)

Particular Values

L(4)L(4) \approx 6.0234612396.023461239
L(12)L(\frac12) \approx 6.0234612396.023461239
L(92)L(\frac{9}{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
5 1p3T 1 - p^{3} T
7 1+p3T 1 + p^{3} T
good3 131pT+p7T2 1 - 31 p T + p^{7} T^{2}
11 1197pT+p7T2 1 - 197 p T + p^{7} T^{2}
13 1+1661T+p7T2 1 + 1661 T + p^{7} T^{2}
17 1+35771T+p7T2 1 + 35771 T + p^{7} T^{2}
19 1+20222T+p7T2 1 + 20222 T + p^{7} T^{2}
23 142130T+p7T2 1 - 42130 T + p^{7} T^{2}
29 1+111789T+p7T2 1 + 111789 T + p^{7} T^{2}
31 1269504T+p7T2 1 - 269504 T + p^{7} T^{2}
37 1532774T+p7T2 1 - 532774 T + p^{7} T^{2}
41 1158056T+p7T2 1 - 158056 T + p^{7} T^{2}
43 1521874T+p7T2 1 - 521874 T + p^{7} T^{2}
47 1939733T+p7T2 1 - 939733 T + p^{7} T^{2}
53 1+408384T+p7T2 1 + 408384 T + p^{7} T^{2}
59 1522172T+p7T2 1 - 522172 T + p^{7} T^{2}
61 1350080T+p7T2 1 - 350080 T + p^{7} T^{2}
67 13931176T+p7T2 1 - 3931176 T + p^{7} T^{2}
71 1+1194016T+p7T2 1 + 1194016 T + p^{7} T^{2}
73 1998350T+p7T2 1 - 998350 T + p^{7} T^{2}
79 12120709T+p7T2 1 - 2120709 T + p^{7} T^{2}
83 11746708T+p7T2 1 - 1746708 T + p^{7} T^{2}
89 1+10077740T+p7T2 1 + 10077740 T + p^{7} T^{2}
97 1+6238295T+p7T2 1 + 6238295 T + p^{7} 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

−9.310856142700045964390073704600, −8.938784345521179340593204862736, −8.047947384715294617251081455186, −7.06989129710357733395268937244, −6.34828391652912194456009179766, −4.55782253145449557394573063654, −3.91989989836331594225487743865, −2.65823716486276100911316390675, −2.23930582823280443556232223492, −0.984664456895481641452636411850, 0.984664456895481641452636411850, 2.23930582823280443556232223492, 2.65823716486276100911316390675, 3.91989989836331594225487743865, 4.55782253145449557394573063654, 6.34828391652912194456009179766, 7.06989129710357733395268937244, 8.047947384715294617251081455186, 8.938784345521179340593204862736, 9.310856142700045964390073704600

Graph of the ZZ-function along the critical line