Properties

Label 2-40425-1.1-c1-0-16
Degree $2$
Conductor $40425$
Sign $1$
Analytic cond. $322.795$
Root an. cond. $17.9665$
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  + 3-s − 2·4-s + 9-s + 11-s − 2·12-s + 2·13-s + 4·16-s + 4·19-s − 6·23-s + 27-s − 5·31-s + 33-s − 2·36-s − 11·37-s + 2·39-s − 6·41-s + 43-s − 2·44-s + 6·47-s + 4·48-s − 4·52-s − 12·53-s + 4·57-s + 6·59-s − 5·61-s − 8·64-s + 4·67-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1/3·9-s + 0.301·11-s − 0.577·12-s + 0.554·13-s + 16-s + 0.917·19-s − 1.25·23-s + 0.192·27-s − 0.898·31-s + 0.174·33-s − 1/3·36-s − 1.80·37-s + 0.320·39-s − 0.937·41-s + 0.152·43-s − 0.301·44-s + 0.875·47-s + 0.577·48-s − 0.554·52-s − 1.64·53-s + 0.529·57-s + 0.781·59-s − 0.640·61-s − 64-s + 0.488·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 40425 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(40425\)    =    \(3 \cdot 5^{2} \cdot 7^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(322.795\)
Root analytic conductor: \(17.9665\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 40425,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.879947126\)
\(L(\frac12)\) \(\approx\) \(1.879947126\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - T \)
5 \( 1 \)
7 \( 1 \)
11 \( 1 - T \)
good2 \( 1 + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - T + p T^{2} \)
47 \( 1 - 6 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 5 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 6 T + p T^{2} \)
73 \( 1 - 11 T + p T^{2} \)
79 \( 1 + T + p T^{2} \)
83 \( 1 + 6 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 7 T + p T^{2} \)
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

−14.52104613189965, −14.18343602867605, −13.76607893441307, −13.46863529826235, −12.62446197189915, −12.39824434431761, −11.73559768690527, −11.07320261917922, −10.39697387330446, −9.949234574614128, −9.371023835337035, −9.004863243657706, −8.351016491350469, −8.033197791921007, −7.330090811297471, −6.742863947697453, −5.944256557881730, −5.412252953526347, −4.822072121262208, −4.094092812794484, −3.555220307025097, −3.199270827107699, −2.049448032655484, −1.462679190539751, −0.5018492964541746, 0.5018492964541746, 1.462679190539751, 2.049448032655484, 3.199270827107699, 3.555220307025097, 4.094092812794484, 4.822072121262208, 5.412252953526347, 5.944256557881730, 6.742863947697453, 7.330090811297471, 8.033197791921007, 8.351016491350469, 9.004863243657706, 9.371023835337035, 9.949234574614128, 10.39697387330446, 11.07320261917922, 11.73559768690527, 12.39824434431761, 12.62446197189915, 13.46863529826235, 13.76607893441307, 14.18343602867605, 14.52104613189965

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