Properties

Label 2-56350-1.1-c1-0-42
Degree $2$
Conductor $56350$
Sign $1$
Analytic cond. $449.957$
Root an. cond. $21.2121$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 2·3-s + 4-s + 2·6-s + 8-s + 9-s + 4·11-s + 2·12-s + 16-s + 6·17-s + 18-s + 6·19-s + 4·22-s + 23-s + 2·24-s − 4·27-s + 10·29-s − 4·31-s + 32-s + 8·33-s + 6·34-s + 36-s + 2·37-s + 6·38-s + 10·41-s + 4·43-s + 4·44-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.15·3-s + 1/2·4-s + 0.816·6-s + 0.353·8-s + 1/3·9-s + 1.20·11-s + 0.577·12-s + 1/4·16-s + 1.45·17-s + 0.235·18-s + 1.37·19-s + 0.852·22-s + 0.208·23-s + 0.408·24-s − 0.769·27-s + 1.85·29-s − 0.718·31-s + 0.176·32-s + 1.39·33-s + 1.02·34-s + 1/6·36-s + 0.328·37-s + 0.973·38-s + 1.56·41-s + 0.609·43-s + 0.603·44-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(56350\)    =    \(2 \cdot 5^{2} \cdot 7^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(449.957\)
Root analytic conductor: \(21.2121\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{56350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 56350,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(9.151125455\)
\(L(\frac12)\) \(\approx\) \(9.151125455\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
23 \( 1 - T \)
good3 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 2 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 14 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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

−14.34830232070968, −14.08623708207683, −13.56038207332266, −12.97288521098678, −12.34752664428037, −11.88491255902795, −11.63181682212767, −10.81154951164326, −10.22003155549113, −9.673254044421183, −9.133598387098264, −8.790808033947474, −8.029935840964607, −7.524439998492505, −7.193065322126072, −6.427184608120300, −5.692176542214983, −5.461571422919369, −4.437235398747516, −4.003316826590235, −3.431358505070903, −2.842224053606056, −2.464369528703508, −1.331642136140156, −0.9968327301178609, 0.9968327301178609, 1.331642136140156, 2.464369528703508, 2.842224053606056, 3.431358505070903, 4.003316826590235, 4.437235398747516, 5.461571422919369, 5.692176542214983, 6.427184608120300, 7.193065322126072, 7.524439998492505, 8.029935840964607, 8.790808033947474, 9.133598387098264, 9.673254044421183, 10.22003155549113, 10.81154951164326, 11.63181682212767, 11.88491255902795, 12.34752664428037, 12.97288521098678, 13.56038207332266, 14.08623708207683, 14.34830232070968

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