↑

Properties

Label 2-303450-1.1-c1-0-24
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 + 2·13-s − 14-s + 16-s − 18-s + 4·19-s − 21-s + 4·22-s + 8·23-s + 24-s − 2·26-s − 27-s + 28-s − 6·29-s − 32-s + 4·33-s + 36-s − 2·37-s − 4·38-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 + 0.554·13-s − 0.267·14-s + 1/4·16-s − 0.235·18-s + 0.917·19-s − 0.218·21-s + 0.852·22-s + 1.66·23-s + 0.204·24-s − 0.392·26-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 − 0.328·37-s − 0.648·38-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

\(L(1)\) \(\approx\) \(1.089112099\)
\(L(\frac12)\) \(\approx\) \(1.089112099\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
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

−12.76673446532563, −12.03157108367868, −11.71537041933185, −11.06842213243002, −10.85795358412580, −10.60769359910814, −9.881034004589973, −9.485929719378236, −9.095540422455080, −8.411540633297985, −8.118005543176635, −7.530404742087164, −7.173372175275025, −6.724115555981320, −6.097671164160960, −5.537742459562110, −5.139637781896093, −4.842670825532602, −4.000775071827662, −3.311590813034024, −2.975448167826479, −2.194675624290670, −1.603887916926693, −1.036549682221809, −0.3650988893623853, 0.3650988893623853, 1.036549682221809, 1.603887916926693, 2.194675624290670, 2.975448167826479, 3.311590813034024, 4.000775071827662, 4.842670825532602, 5.139637781896093, 5.537742459562110, 6.097671164160960, 6.724115555981320, 7.173372175275025, 7.530404742087164, 8.118005543176635, 8.411540633297985, 9.095540422455080, 9.485929719378236, 9.881034004589973, 10.60769359910814, 10.85795358412580, 11.06842213243002, 11.71537041933185, 12.03157108367868, 12.76673446532563

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