Properties

Label 2-236992-1.1-c1-0-30
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 − 2·5-s − 7-s + 9-s + 6·11-s + 4·13-s + 4·15-s + 2·17-s + 4·19-s + 2·21-s − 25-s + 4·27-s + 10·29-s − 8·31-s − 12·33-s + 2·35-s − 8·37-s − 8·39-s − 2·41-s + 6·43-s − 2·45-s + 12·47-s + 49-s − 4·51-s + 12·53-s − 12·55-s − 8·57-s + ⋯
L(s)  = 1  − 1.15·3-s − 0.894·5-s − 0.377·7-s + 1/3·9-s + 1.80·11-s + 1.10·13-s + 1.03·15-s + 0.485·17-s + 0.917·19-s + 0.436·21-s − 1/5·25-s + 0.769·27-s + 1.85·29-s − 1.43·31-s − 2.08·33-s + 0.338·35-s − 1.31·37-s − 1.28·39-s − 0.312·41-s + 0.914·43-s − 0.298·45-s + 1.75·47-s + 1/7·49-s − 0.560·51-s + 1.64·53-s − 1.61·55-s − 1.05·57-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

\(L(1)\) \(\approx\) \(1.999773477\)
\(L(\frac12)\) \(\approx\) \(1.999773477\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
11 \( 1 - 6 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 10 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 12 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 2 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
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

−12.43933296644727, −12.28851504213891, −12.06662006777448, −11.50760974668501, −11.19282408022445, −10.74781415428832, −10.22920431030122, −9.718891668596203, −9.055803507946016, −8.705179784237032, −8.363602527373006, −7.422816283333454, −7.223183488672443, −6.638329654266817, −6.177979416331654, −5.810764335969586, −5.260833931296014, −4.694847378946508, −3.929510088800681, −3.748258251881773, −3.302518243987582, −2.409440187900618, −1.481018668298016, −0.8972195262610782, −0.5807224724567402, 0.5807224724567402, 0.8972195262610782, 1.481018668298016, 2.409440187900618, 3.302518243987582, 3.748258251881773, 3.929510088800681, 4.694847378946508, 5.260833931296014, 5.810764335969586, 6.177979416331654, 6.638329654266817, 7.223183488672443, 7.422816283333454, 8.363602527373006, 8.705179784237032, 9.055803507946016, 9.718891668596203, 10.22920431030122, 10.74781415428832, 11.19282408022445, 11.50760974668501, 12.06662006777448, 12.28851504213891, 12.43933296644727

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