Properties

Label 2-190575-1.1-c1-0-20
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

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 2·4-s + 7-s − 2·13-s + 2·14-s − 4·16-s + 3·17-s − 19-s − 7·23-s − 4·26-s + 2·28-s + 29-s + 8·31-s − 8·32-s + 6·34-s + 2·37-s − 2·38-s − 8·41-s + 9·43-s − 14·46-s − 12·47-s + 49-s − 4·52-s − 5·53-s + 2·58-s − 3·59-s − 13·61-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.377·7-s − 0.554·13-s + 0.534·14-s − 16-s + 0.727·17-s − 0.229·19-s − 1.45·23-s − 0.784·26-s + 0.377·28-s + 0.185·29-s + 1.43·31-s − 1.41·32-s + 1.02·34-s + 0.328·37-s − 0.324·38-s − 1.24·41-s + 1.37·43-s − 2.06·46-s − 1.75·47-s + 1/7·49-s − 0.554·52-s − 0.686·53-s + 0.262·58-s − 0.390·59-s − 1.66·61-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: \(0\)
Selberg data: \((2,\ 190575,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.722698478\)
\(L(\frac12)\) \(\approx\) \(3.722698478\)
\(L(\frac{3}{2})\) 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 \( 1 + 2 T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + T + p T^{2} \)
23 \( 1 + 7 T + p T^{2} \)
29 \( 1 - T + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 9 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 + 3 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 14 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 5 T + p T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.10185617998214, −12.61622386965666, −12.26064784790399, −11.85177600949992, −11.49709324413439, −10.95683993762652, −10.25610004134173, −9.960950537735426, −9.396635760254854, −8.806664369205571, −8.108485643002276, −7.892369505468580, −7.240861007433589, −6.543259454067675, −6.235823973703329, −5.741236681686657, −5.158830611356233, −4.676088224548770, −4.356003178004606, −3.717588701917745, −3.157253109273279, −2.676979225877844, −2.027624841623404, −1.426500869553128, −0.4059398173503711, 0.4059398173503711, 1.426500869553128, 2.027624841623404, 2.676979225877844, 3.157253109273279, 3.717588701917745, 4.356003178004606, 4.676088224548770, 5.158830611356233, 5.741236681686657, 6.235823973703329, 6.543259454067675, 7.240861007433589, 7.892369505468580, 8.108485643002276, 8.806664369205571, 9.396635760254854, 9.960950537735426, 10.25610004134173, 10.95683993762652, 11.49709324413439, 11.85177600949992, 12.26064784790399, 12.61622386965666, 13.10185617998214

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