Properties

Label 2-236992-1.1-c1-0-2
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  − 7-s − 3·9-s − 4·11-s + 2·13-s − 4·17-s + 4·19-s − 5·25-s − 2·29-s + 4·31-s + 4·37-s − 6·41-s − 4·43-s + 4·47-s + 49-s + 12·53-s − 8·59-s + 3·63-s + 4·67-s + 8·71-s − 10·73-s + 4·77-s + 8·79-s + 9·81-s − 12·83-s + 4·89-s − 2·91-s − 12·97-s + ⋯
L(s)  = 1  − 0.377·7-s − 9-s − 1.20·11-s + 0.554·13-s − 0.970·17-s + 0.917·19-s − 25-s − 0.371·29-s + 0.718·31-s + 0.657·37-s − 0.937·41-s − 0.609·43-s + 0.583·47-s + 1/7·49-s + 1.64·53-s − 1.04·59-s + 0.377·63-s + 0.488·67-s + 0.949·71-s − 1.17·73-s + 0.455·77-s + 0.900·79-s + 81-s − 1.31·83-s + 0.423·89-s − 0.209·91-s − 1.21·97-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

\(L(1)\) \(\approx\) \(0.5531191159\)
\(L(\frac12)\) \(\approx\) \(0.5531191159\)
\(L(\frac{3}{2})\) 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 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 - 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 - 4 T + p T^{2} \)
53 \( 1 - 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 - 4 T + p T^{2} \)
97 \( 1 + 12 T + p T^{2} \)
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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

−13.11180817932771, −12.36290490864253, −11.93319039941752, −11.51513688491297, −11.04492936898828, −10.67603396897965, −10.04725882793552, −9.738911911023359, −9.101640355749061, −8.653590887706426, −8.247608009799448, −7.735726196876270, −7.317369772912337, −6.607361415401402, −6.218840856536533, −5.636004038136136, −5.294341363555433, −4.776263306217033, −3.974250627914384, −3.620115999902197, −2.757837769418382, −2.664176532144676, −1.898696818985704, −1.069039661021805, −0.2172691447759368, 0.2172691447759368, 1.069039661021805, 1.898696818985704, 2.664176532144676, 2.757837769418382, 3.620115999902197, 3.974250627914384, 4.776263306217033, 5.294341363555433, 5.636004038136136, 6.218840856536533, 6.607361415401402, 7.317369772912337, 7.735726196876270, 8.247608009799448, 8.653590887706426, 9.101640355749061, 9.738911911023359, 10.04725882793552, 10.67603396897965, 11.04492936898828, 11.51513688491297, 11.93319039941752, 12.36290490864253, 13.11180817932771

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