Properties

Label 2-190575-1.1-c1-0-1
Degree $2$
Conductor $190575$
Sign $1$
Analytic cond. $1521.74$
Root an. cond. $39.0096$
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-s − 4-s + 7-s − 3·8-s − 2·13-s + 14-s − 16-s − 2·17-s − 4·19-s − 2·26-s − 28-s + 6·29-s + 5·32-s − 2·34-s − 6·37-s − 4·38-s − 6·41-s − 4·43-s + 49-s + 2·52-s − 2·53-s − 3·56-s + 6·58-s − 4·59-s − 6·61-s + 7·64-s − 12·67-s + ⋯
L(s)  = 1  + 0.707·2-s − 1/2·4-s + 0.377·7-s − 1.06·8-s − 0.554·13-s + 0.267·14-s − 1/4·16-s − 0.485·17-s − 0.917·19-s − 0.392·26-s − 0.188·28-s + 1.11·29-s + 0.883·32-s − 0.342·34-s − 0.986·37-s − 0.648·38-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.277·52-s − 0.274·53-s − 0.400·56-s + 0.787·58-s − 0.520·59-s − 0.768·61-s + 7/8·64-s − 1.46·67-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 190575 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(190575\)    =    \(3^{2} \cdot 5^{2} \cdot 7 \cdot 11^{2}\)
Sign: $1$
Analytic conductor: \(1521.74\)
Root analytic conductor: \(39.0096\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{190575} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 190575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5692343220\)
\(L(\frac12)\) \(\approx\) \(0.5692343220\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 \)
good2 \( 1 - T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 2 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 6 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 10 T + p T^{2} \)
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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

−13.29319553899973, −12.46077554794384, −12.35000145621259, −11.94012798536155, −11.27811252460345, −10.81925194098739, −10.25937259150522, −9.872144955867895, −9.247060241945792, −8.807844750287872, −8.344509951102485, −8.017399540156727, −7.210179125053364, −6.723421133966218, −6.289262588013212, −5.700856082599447, −5.090036162974811, −4.777479332062149, −4.280660233411301, −3.798903419450861, −3.068156874708034, −2.666109985472084, −1.897510851959664, −1.254647851246648, −0.1910694254212603, 0.1910694254212603, 1.254647851246648, 1.897510851959664, 2.666109985472084, 3.068156874708034, 3.798903419450861, 4.280660233411301, 4.777479332062149, 5.090036162974811, 5.700856082599447, 6.289262588013212, 6.723421133966218, 7.210179125053364, 8.017399540156727, 8.344509951102485, 8.807844750287872, 9.247060241945792, 9.872144955867895, 10.25937259150522, 10.81925194098739, 11.27811252460345, 11.94012798536155, 12.35000145621259, 12.46077554794384, 13.29319553899973

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