Properties

Label 2-250173-1.1-c1-0-15
Degree $2$
Conductor $250173$
Sign $-1$
Analytic cond. $1997.64$
Root an. cond. $44.6949$
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  − 2·4-s − 3·5-s + 7-s − 11-s + 4·13-s + 4·16-s + 6·20-s − 6·23-s + 4·25-s − 2·28-s + 6·29-s − 5·31-s − 3·35-s − 8·37-s − 9·41-s + 2·43-s + 2·44-s − 9·47-s + 49-s − 8·52-s − 3·53-s + 3·55-s − 7·61-s − 8·64-s − 12·65-s + 4·67-s − 9·71-s + ⋯
L(s)  = 1  − 4-s − 1.34·5-s + 0.377·7-s − 0.301·11-s + 1.10·13-s + 16-s + 1.34·20-s − 1.25·23-s + 4/5·25-s − 0.377·28-s + 1.11·29-s − 0.898·31-s − 0.507·35-s − 1.31·37-s − 1.40·41-s + 0.304·43-s + 0.301·44-s − 1.31·47-s + 1/7·49-s − 1.10·52-s − 0.412·53-s + 0.404·55-s − 0.896·61-s − 64-s − 1.48·65-s + 0.488·67-s − 1.06·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 250173 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(250173\)    =    \(3^{2} \cdot 7 \cdot 11 \cdot 19^{2}\)
Sign: $-1$
Analytic conductor: \(1997.64\)
Root analytic conductor: \(44.6949\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 250173,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
19 \( 1 \)
good2 \( 1 + p T^{2} \)
5 \( 1 + 3 T + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 + 8 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 2 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 11 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−13.14392513615330, −12.49627704045474, −12.22099121098112, −11.69973189773500, −11.34241065634121, −10.76220481393329, −10.27430028848973, −9.994110990033475, −9.152517176051775, −8.812423476726031, −8.325922676130418, −8.002816509422715, −7.732252557611551, −6.944400147714306, −6.532163379737751, −5.769495793188626, −5.387704742492362, −4.744052936730951, −4.322797298200779, −3.917158200692196, −3.318751771419579, −3.085855881498889, −1.879325010789506, −1.441237369104189, −0.5483932096233678, 0, 0.5483932096233678, 1.441237369104189, 1.879325010789506, 3.085855881498889, 3.318751771419579, 3.917158200692196, 4.322797298200779, 4.744052936730951, 5.387704742492362, 5.769495793188626, 6.532163379737751, 6.944400147714306, 7.732252557611551, 8.002816509422715, 8.325922676130418, 8.812423476726031, 9.152517176051775, 9.994110990033475, 10.27430028848973, 10.76220481393329, 11.34241065634121, 11.69973189773500, 12.22099121098112, 12.49627704045474, 13.14392513615330

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