Properties

Label 2-190575-1.1-c1-0-116
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2·2-s + 2·4-s − 7-s + 7·13-s + 2·14-s − 4·16-s − 3·17-s + 3·19-s + 8·23-s − 14·26-s − 2·28-s − 4·31-s + 8·32-s + 6·34-s − 3·37-s − 6·38-s − 3·41-s − 6·43-s − 16·46-s + 10·47-s + 49-s + 14·52-s − 3·53-s + 6·59-s + 13·61-s + 8·62-s − 8·64-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s − 0.377·7-s + 1.94·13-s + 0.534·14-s − 16-s − 0.727·17-s + 0.688·19-s + 1.66·23-s − 2.74·26-s − 0.377·28-s − 0.718·31-s + 1.41·32-s + 1.02·34-s − 0.493·37-s − 0.973·38-s − 0.468·41-s − 0.914·43-s − 2.35·46-s + 1.45·47-s + 1/7·49-s + 1.94·52-s − 0.412·53-s + 0.781·59-s + 1.66·61-s + 1.01·62-s − 64-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 190575,\ (\ :1/2),\ -1)\)

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 \( 1 \)
5 \( 1 \)
7 \( 1 + T \)
11 \( 1 \)
good2 \( 1 + p T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 - 3 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 3 T + p T^{2} \)
41 \( 1 + 3 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + 3 T + p T^{2} \)
59 \( 1 - 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 3 T + p T^{2} \)
71 \( 1 - 13 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 4 T + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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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

−13.38841071087633, −12.92646080770368, −12.40110575366397, −11.59235176128337, −11.23375111922668, −11.01165741549291, −10.36698509468281, −10.13621786853850, −9.278197607444589, −9.099902539242601, −8.736193847840150, −8.243654278648720, −7.750806121626853, −7.118263880627592, −6.668579919755568, −6.450365278262032, −5.521805871040862, −5.210612040294900, −4.410869404038891, −3.717646424078547, −3.386834955093960, −2.569639983013691, −1.950960126408884, −1.175714543617861, −0.9046038754159338, 0, 0.9046038754159338, 1.175714543617861, 1.950960126408884, 2.569639983013691, 3.386834955093960, 3.717646424078547, 4.410869404038891, 5.210612040294900, 5.521805871040862, 6.450365278262032, 6.668579919755568, 7.118263880627592, 7.750806121626853, 8.243654278648720, 8.736193847840150, 9.099902539242601, 9.278197607444589, 10.13621786853850, 10.36698509468281, 11.01165741549291, 11.23375111922668, 11.59235176128337, 12.40110575366397, 12.92646080770368, 13.38841071087633

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