Properties

Label 2-236992-1.1-c1-0-36
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2·3-s − 4·5-s − 7-s + 9-s − 4·11-s − 4·13-s − 8·15-s + 6·17-s − 2·19-s − 2·21-s + 11·25-s − 4·27-s + 6·29-s + 8·31-s − 8·33-s + 4·35-s + 10·37-s − 8·39-s + 6·41-s − 12·43-s − 4·45-s + 8·47-s + 49-s + 12·51-s − 2·53-s + 16·55-s − 4·57-s + ⋯
L(s)  = 1  + 1.15·3-s − 1.78·5-s − 0.377·7-s + 1/3·9-s − 1.20·11-s − 1.10·13-s − 2.06·15-s + 1.45·17-s − 0.458·19-s − 0.436·21-s + 11/5·25-s − 0.769·27-s + 1.11·29-s + 1.43·31-s − 1.39·33-s + 0.676·35-s + 1.64·37-s − 1.28·39-s + 0.937·41-s − 1.82·43-s − 0.596·45-s + 1.16·47-s + 1/7·49-s + 1.68·51-s − 0.274·53-s + 2.15·55-s − 0.529·57-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: \(1\)
Selberg data: \((2,\ 236992,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 10 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 14 T + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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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

−13.05034187786706, −12.65475575725350, −12.19397481611998, −11.85000687834056, −11.45324661149961, −10.68601684821506, −10.31432042079880, −9.904042238455128, −9.347101296917787, −8.772733729345662, −8.264874941106785, −7.901872590991837, −7.683503912637257, −7.328884057809247, −6.614341754899958, −6.005208617166620, −5.303261125647722, −4.684305184121362, −4.352857596751673, −3.781657526472783, −3.072267010178960, −2.782753404692775, −2.617147184435778, −1.451078606800554, −0.6354340815375321, 0, 0.6354340815375321, 1.451078606800554, 2.617147184435778, 2.782753404692775, 3.072267010178960, 3.781657526472783, 4.352857596751673, 4.684305184121362, 5.303261125647722, 6.005208617166620, 6.614341754899958, 7.328884057809247, 7.683503912637257, 7.901872590991837, 8.264874941106785, 8.772733729345662, 9.347101296917787, 9.904042238455128, 10.31432042079880, 10.68601684821506, 11.45324661149961, 11.85000687834056, 12.19397481611998, 12.65475575725350, 13.05034187786706

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