Properties

Label 2-236992-1.1-c1-0-11
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  + 2·5-s − 7-s − 3·9-s + 4·11-s − 6·13-s + 2·17-s + 4·19-s − 25-s + 2·29-s − 4·31-s − 2·35-s − 2·37-s − 6·41-s + 12·43-s − 6·45-s − 12·47-s + 49-s − 10·53-s + 8·55-s + 2·61-s + 3·63-s − 12·65-s + 12·67-s + 8·71-s − 14·73-s − 4·77-s − 8·79-s + ⋯
L(s)  = 1  + 0.894·5-s − 0.377·7-s − 9-s + 1.20·11-s − 1.66·13-s + 0.485·17-s + 0.917·19-s − 1/5·25-s + 0.371·29-s − 0.718·31-s − 0.338·35-s − 0.328·37-s − 0.937·41-s + 1.82·43-s − 0.894·45-s − 1.75·47-s + 1/7·49-s − 1.37·53-s + 1.07·55-s + 0.256·61-s + 0.377·63-s − 1.48·65-s + 1.46·67-s + 0.949·71-s − 1.63·73-s − 0.455·77-s − 0.900·79-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

\(L(1)\) \(\approx\) \(1.688636025\)
\(L(\frac12)\) \(\approx\) \(1.688636025\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 6 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 - 12 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 10 T + p T^{2} \)
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

−12.91825848941159, −12.27860645518844, −12.09676357107397, −11.53898747792040, −11.16411153949127, −10.51536346924373, −9.813873733216337, −9.703452720327309, −9.350311729462084, −8.816623193289655, −8.204467226318208, −7.717878413996367, −7.079136771173218, −6.762421143888265, −6.150797179682870, −5.668429177898922, −5.323834840150342, −4.764263835176793, −4.143089519980127, −3.337899288168594, −3.084697256510049, −2.380963620987256, −1.858136350464254, −1.215197543941265, −0.3555141082161409, 0.3555141082161409, 1.215197543941265, 1.858136350464254, 2.380963620987256, 3.084697256510049, 3.337899288168594, 4.143089519980127, 4.764263835176793, 5.323834840150342, 5.668429177898922, 6.150797179682870, 6.762421143888265, 7.079136771173218, 7.717878413996367, 8.204467226318208, 8.816623193289655, 9.350311729462084, 9.703452720327309, 9.813873733216337, 10.51536346924373, 11.16411153949127, 11.53898747792040, 12.09676357107397, 12.27860645518844, 12.91825848941159

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