Properties

Label 2-236992-1.1-c1-0-44
Degree $2$
Conductor $236992$
Sign $1$
Analytic cond. $1892.39$
Root an. cond. $43.5016$
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  + 2·3-s + 4·5-s + 7-s + 9-s − 4·11-s + 2·13-s + 8·15-s + 4·17-s + 2·21-s + 11·25-s − 4·27-s − 2·29-s + 6·31-s − 8·33-s + 4·35-s + 2·37-s + 4·39-s − 6·41-s + 4·43-s + 4·45-s + 10·47-s + 49-s + 8·51-s + 6·53-s − 16·55-s − 2·59-s + 8·61-s + ⋯
L(s)  = 1  + 1.15·3-s + 1.78·5-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 0.554·13-s + 2.06·15-s + 0.970·17-s + 0.436·21-s + 11/5·25-s − 0.769·27-s − 0.371·29-s + 1.07·31-s − 1.39·33-s + 0.676·35-s + 0.328·37-s + 0.640·39-s − 0.937·41-s + 0.609·43-s + 0.596·45-s + 1.45·47-s + 1/7·49-s + 1.12·51-s + 0.824·53-s − 2.15·55-s − 0.260·59-s + 1.02·61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(236992\)    =    \(2^{6} \cdot 7 \cdot 23^{2}\)
Sign: $1$
Analytic conductor: \(1892.39\)
Root analytic conductor: \(43.5016\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{236992} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(8.030290936\)
\(L(\frac12)\) \(\approx\) \(8.030290936\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
23 \( 1 \)
good3 \( 1 - 2 T + p T^{2} \)
5 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 6 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 2 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−13.02032873396729, −12.76105231906669, −12.05706031361270, −11.49266786408184, −10.78738978563893, −10.51131363992201, −9.979783064859620, −9.650331484318714, −9.226508841080519, −8.587322381932145, −8.365995019376686, −7.835002580847693, −7.324860163121497, −6.736383845738011, −6.058946670033123, −5.702217639355876, −5.256065660003314, −4.838896883042122, −3.942447856403100, −3.461317660044979, −2.702720475962380, −2.490301486839100, −2.011802199372651, −1.318579733528392, −0.7157079764298335, 0.7157079764298335, 1.318579733528392, 2.011802199372651, 2.490301486839100, 2.702720475962380, 3.461317660044979, 3.942447856403100, 4.838896883042122, 5.256065660003314, 5.702217639355876, 6.058946670033123, 6.736383845738011, 7.324860163121497, 7.835002580847693, 8.365995019376686, 8.587322381932145, 9.226508841080519, 9.650331484318714, 9.979783064859620, 10.51131363992201, 10.78738978563893, 11.49266786408184, 12.05706031361270, 12.76105231906669, 13.02032873396729

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