↑

Properties

Label 2-1785-1.1-c1-0-11
Degree 22
Conductor 17851785
Sign 11
Analytic cond. 14.253214.2532
Root an. cond. 3.775353.77535
Motivic weight 11
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank 00

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 3-s − 4-s + 5-s − 6-s − 7-s − 3·8-s + 9-s + 10-s + 12-s − 14-s − 15-s − 16-s − 17-s + 18-s + 2·19-s − 20-s + 21-s + 6·23-s + 3·24-s + 25-s − 27-s + 28-s − 4·29-s − 30-s − 4·31-s + 5·32-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s − 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.377·7-s − 1.06·8-s + 1/3·9-s + 0.316·10-s + 0.288·12-s − 0.267·14-s − 0.258·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s + 0.458·19-s − 0.223·20-s + 0.218·21-s + 1.25·23-s + 0.612·24-s + 1/5·25-s − 0.192·27-s + 0.188·28-s − 0.742·29-s − 0.182·30-s − 0.718·31-s + 0.883·32-s + ⋯

Functional equation

Λ(s)=(1785s/2 ΓC(s) L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1785s/2 ΓC(s+1/2) L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1785 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 22
Conductor: 17851785    =    3⋅5⋅7⋅173 \cdot 5 \cdot 7 \cdot 17
Sign: 11
Analytic conductor: 14.253214.2532
Root analytic conductor: 3.775353.77535
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: 00
Selberg data: (2, 1785, ( :1/2), 1)(2,\ 1785,\ (\ :1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 1.6093579951.609357995
L(12)L(\frac12) ≈\approx 1.6093579951.609357995
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)
bad3 1+T 1 + T
5 1−T 1 - T
7 1+T 1 + T
17 1+T 1 + T
good2 1−T+pT2 1 - T + p T^{2}
11 1+pT2 1 + p T^{2}
13 1+pT2 1 + p T^{2}
19 1−2T+pT2 1 - 2 T + p T^{2}
23 1−6T+pT2 1 - 6 T + p T^{2}
29 1+4T+pT2 1 + 4 T + p T^{2}
31 1+4T+pT2 1 + 4 T + p T^{2}
37 1−4T+pT2 1 - 4 T + p T^{2}
41 1−10T+pT2 1 - 10 T + p T^{2}
43 1−4T+pT2 1 - 4 T + p T^{2}
47 1−6T+pT2 1 - 6 T + p T^{2}
53 1+10T+pT2 1 + 10 T + p T^{2}
59 1−12T+pT2 1 - 12 T + p T^{2}
61 1−14T+pT2 1 - 14 T + p T^{2}
67 1−12T+pT2 1 - 12 T + p T^{2}
71 1+pT2 1 + p T^{2}
73 1+10T+pT2 1 + 10 T + p T^{2}
79 1−14T+pT2 1 - 14 T + p T^{2}
83 1+18T+pT2 1 + 18 T + p T^{2}
89 1+pT2 1 + p T^{2}
97 1−2T+pT2 1 - 2 T + p T^{2}
show more
show less
   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

−9.374382510909813445824217205848, −8.702357198738553963201149941811, −7.50835306182715923625670381327, −6.67931663262518992670154334973, −5.79530321478153998743024153736, −5.32363110257955187072323371443, −4.40788901588214039441553602026, −3.55589293847504590895677387951, −2.47789863858984368696646328047, −0.809464007514526215837722664467, 0.809464007514526215837722664467, 2.47789863858984368696646328047, 3.55589293847504590895677387951, 4.40788901588214039441553602026, 5.32363110257955187072323371443, 5.79530321478153998743024153736, 6.67931663262518992670154334973, 7.50835306182715923625670381327, 8.702357198738553963201149941811, 9.374382510909813445824217205848

Graph of the ZZ-function along the critical line