Properties

Label 2-8450-1.1-c1-0-180
Degree $2$
Conductor $8450$
Sign $1$
Analytic cond. $67.4735$
Root an. cond. $8.21423$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 3·3-s + 4-s + 3·6-s + 7-s + 8-s + 6·9-s + 2·11-s + 3·12-s + 14-s + 16-s + 3·17-s + 6·18-s − 6·19-s + 3·21-s + 2·22-s + 4·23-s + 3·24-s + 9·27-s + 28-s + 2·29-s − 4·31-s + 32-s + 6·33-s + 3·34-s + 6·36-s + 3·37-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.377·7-s + 0.353·8-s + 2·9-s + 0.603·11-s + 0.866·12-s + 0.267·14-s + 1/4·16-s + 0.727·17-s + 1.41·18-s − 1.37·19-s + 0.654·21-s + 0.426·22-s + 0.834·23-s + 0.612·24-s + 1.73·27-s + 0.188·28-s + 0.371·29-s − 0.718·31-s + 0.176·32-s + 1.04·33-s + 0.514·34-s + 36-s + 0.493·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8450\)    =    \(2 \cdot 5^{2} \cdot 13^{2}\)
Sign: $1$
Analytic conductor: \(67.4735\)
Root analytic conductor: \(8.21423\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(7.548084687\)
\(L(\frac12)\) \(\approx\) \(7.548084687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 - T \)
5 \( 1 \)
13 \( 1 \)
good3 \( 1 - p T + p T^{2} \) 1.3.ad
7 \( 1 - T + p T^{2} \) 1.7.ab
11 \( 1 - 2 T + p T^{2} \) 1.11.ac
17 \( 1 - 3 T + p T^{2} \) 1.17.ad
19 \( 1 + 6 T + p T^{2} \) 1.19.g
23 \( 1 - 4 T + p T^{2} \) 1.23.ae
29 \( 1 - 2 T + p T^{2} \) 1.29.ac
31 \( 1 + 4 T + p T^{2} \) 1.31.e
37 \( 1 - 3 T + p T^{2} \) 1.37.ad
41 \( 1 + p T^{2} \) 1.41.a
43 \( 1 - 5 T + p T^{2} \) 1.43.af
47 \( 1 - 13 T + p T^{2} \) 1.47.an
53 \( 1 + 12 T + p T^{2} \) 1.53.m
59 \( 1 - 10 T + p T^{2} \) 1.59.ak
61 \( 1 + 8 T + p T^{2} \) 1.61.i
67 \( 1 + 2 T + p T^{2} \) 1.67.c
71 \( 1 - 5 T + p T^{2} \) 1.71.af
73 \( 1 + 10 T + p T^{2} \) 1.73.k
79 \( 1 + 4 T + p T^{2} \) 1.79.e
83 \( 1 + p T^{2} \) 1.83.a
89 \( 1 + 6 T + p T^{2} \) 1.89.g
97 \( 1 - 14 T + p T^{2} \) 1.97.ao
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−7.78618160341898961599148312743, −7.23568791400333437059684221061, −6.51632105010820945514209338703, −5.69499349255850374306805786509, −4.65916643268043458244706899373, −4.15748343893950010861474305754, −3.45971551695615743734182930904, −2.76910930427420910776332481273, −2.04502161572044176240278653055, −1.23311319217689188134168770571, 1.23311319217689188134168770571, 2.04502161572044176240278653055, 2.76910930427420910776332481273, 3.45971551695615743734182930904, 4.15748343893950010861474305754, 4.65916643268043458244706899373, 5.69499349255850374306805786509, 6.51632105010820945514209338703, 7.23568791400333437059684221061, 7.78618160341898961599148312743

Graph of the $Z$-function along the critical line