Properties

Label 2-236992-1.1-c1-0-10
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s + 7-s + 9-s − 4·11-s + 4·13-s + 2·17-s − 6·19-s + 2·21-s − 5·25-s − 4·27-s − 2·29-s − 4·31-s − 8·33-s + 10·37-s + 8·39-s − 10·41-s + 4·43-s + 4·47-s + 49-s + 4·51-s − 2·53-s − 12·57-s − 10·59-s − 8·61-s + 63-s − 8·67-s − 6·73-s + ⋯
L(s)  = 1  + 1.15·3-s + 0.377·7-s + 1/3·9-s − 1.20·11-s + 1.10·13-s + 0.485·17-s − 1.37·19-s + 0.436·21-s − 25-s − 0.769·27-s − 0.371·29-s − 0.718·31-s − 1.39·33-s + 1.64·37-s + 1.28·39-s − 1.56·41-s + 0.609·43-s + 0.583·47-s + 1/7·49-s + 0.560·51-s − 0.274·53-s − 1.58·57-s − 1.30·59-s − 1.02·61-s + 0.125·63-s − 0.977·67-s − 0.702·73-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

\(L(1)\) \(\approx\) \(2.025874595\)
\(L(\frac12)\) \(\approx\) \(2.025874595\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 18 T + p T^{2} \)
97 \( 1 - 2 T + p T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.10538504700834, −12.56588706118657, −12.05063050051957, −11.40339317328516, −10.97847143709948, −10.58141458073575, −10.17554533201752, −9.404527170870173, −9.199273422656885, −8.585223177132431, −8.142198430445182, −7.883942041417186, −7.502237678727183, −6.806019287673297, −5.977174630065482, −5.898817955511504, −5.196792518742048, −4.473242290197708, −4.065730834436870, −3.493135133143293, −2.974674524561930, −2.454165471222341, −1.876375417950735, −1.421621868139960, −0.3354250938591134, 0.3354250938591134, 1.421621868139960, 1.876375417950735, 2.454165471222341, 2.974674524561930, 3.493135133143293, 4.065730834436870, 4.473242290197708, 5.196792518742048, 5.898817955511504, 5.977174630065482, 6.806019287673297, 7.502237678727183, 7.883942041417186, 8.142198430445182, 8.585223177132431, 9.199273422656885, 9.404527170870173, 10.17554533201752, 10.58141458073575, 10.97847143709948, 11.40339317328516, 12.05063050051957, 12.56588706118657, 13.10538504700834

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