↑

Properties

Label 2-303450-1.1-c1-0-101
Degree $2$
Conductor $303450$
Sign $1$
Analytic cond. $2423.06$
Root an. cond. $49.2245$
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 − 3-s + 4-s + 6-s + 7-s − 8-s + 9-s + 4·11-s − 12-s + 6·13-s − 14-s + 16-s − 18-s − 21-s − 4·22-s + 4·23-s + 24-s − 6·26-s − 27-s + 28-s + 6·29-s − 32-s − 4·33-s + 36-s − 6·37-s − 6·39-s + 6·41-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.377·7-s − 0.353·8-s + 1/3·9-s + 1.20·11-s − 0.288·12-s + 1.66·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s − 0.218·21-s − 0.852·22-s + 0.834·23-s + 0.204·24-s − 1.17·26-s − 0.192·27-s + 0.188·28-s + 1.11·29-s − 0.176·32-s − 0.696·33-s + 1/6·36-s − 0.986·37-s − 0.960·39-s + 0.937·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 303450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(303450\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 7 \cdot 17^{2}\)
Sign: $1$
Analytic conductor: \(2423.06\)
Root analytic conductor: \(49.2245\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 303450,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.940296175\)
\(L(\frac12)\) \(\approx\) \(2.940296175\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
3 \( 1 + T \)
5 \( 1 \)
7 \( 1 - T \)
17 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 6 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
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

−12.52305914790671, −12.15081945927743, −11.59744076473972, −11.18184941881705, −10.98030826095864, −10.42138921155912, −10.04782631151996, −9.285769344501451, −8.998066250860117, −8.645986511874206, −8.156558459851843, −7.541077643468403, −7.016382580115652, −6.666516265982611, −6.013389520764228, −5.909244774407152, −5.158224260917230, −4.472066721993334, −4.104156934378936, −3.455129414603198, −2.984357604708847, −2.093610970426559, −1.559357424012636, −0.9302691691684868, −0.7003331141816475, 0.7003331141816475, 0.9302691691684868, 1.559357424012636, 2.093610970426559, 2.984357604708847, 3.455129414603198, 4.104156934378936, 4.472066721993334, 5.158224260917230, 5.909244774407152, 6.013389520764228, 6.666516265982611, 7.016382580115652, 7.541077643468403, 8.156558459851843, 8.645986511874206, 8.998066250860117, 9.285769344501451, 10.04782631151996, 10.42138921155912, 10.98030826095864, 11.18184941881705, 11.59744076473972, 12.15081945927743, 12.52305914790671

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