Properties

Label 2-250173-1.1-c1-0-29
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-s − 4-s + 3·5-s − 7-s + 3·8-s − 3·10-s − 11-s + 2·13-s + 14-s − 16-s − 4·17-s − 3·20-s + 22-s + 3·23-s + 4·25-s − 2·26-s + 28-s − 8·29-s − 2·31-s − 5·32-s + 4·34-s − 3·35-s + 2·37-s + 9·40-s + 9·41-s + 11·43-s + 44-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.34·5-s − 0.377·7-s + 1.06·8-s − 0.948·10-s − 0.301·11-s + 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.970·17-s − 0.670·20-s + 0.213·22-s + 0.625·23-s + 4/5·25-s − 0.392·26-s + 0.188·28-s − 1.48·29-s − 0.359·31-s − 0.883·32-s + 0.685·34-s − 0.507·35-s + 0.328·37-s + 1.42·40-s + 1.40·41-s + 1.67·43-s + 0.150·44-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 \( 1 + T + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 3 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 9 T + p T^{2} \)
43 \( 1 - 11 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 11 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 10 T + p T^{2} \)
73 \( 1 + 13 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 - 2 T + p T^{2} \)
97 \( 1 - 19 T + p T^{2} \)
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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

−13.06247865308167, −12.83631155009023, −12.39310387268508, −11.44983329102912, −10.98505950974328, −10.72055678488926, −10.25022392158402, −9.713311616231831, −9.241593003876055, −8.973961400198871, −8.826163652624938, −7.859702280657012, −7.511507016460823, −7.107006269678360, −6.292282387622571, −5.935374095945476, −5.602414651819871, −4.922553544672705, −4.379673600689298, −3.920028151224568, −3.185376249270979, −2.454186151091864, −2.056838126215557, −1.379405336647101, −0.7925452685285312, 0, 0.7925452685285312, 1.379405336647101, 2.056838126215557, 2.454186151091864, 3.185376249270979, 3.920028151224568, 4.379673600689298, 4.922553544672705, 5.602414651819871, 5.935374095945476, 6.292282387622571, 7.107006269678360, 7.511507016460823, 7.859702280657012, 8.826163652624938, 8.973961400198871, 9.241593003876055, 9.713311616231831, 10.25022392158402, 10.72055678488926, 10.98505950974328, 11.44983329102912, 12.39310387268508, 12.83631155009023, 13.06247865308167

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