Properties

Label 2-229320-1.1-c1-0-47
Degree $2$
Conductor $229320$
Sign $-1$
Analytic cond. $1831.12$
Root an. cond. $42.7916$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 5-s − 6·11-s − 13-s − 4·19-s + 4·23-s + 25-s − 6·29-s − 10·31-s + 10·37-s − 8·41-s − 2·43-s − 12·53-s + 6·55-s − 12·59-s + 10·61-s + 65-s + 14·73-s + 12·83-s + 4·95-s − 6·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.80·11-s − 0.277·13-s − 0.917·19-s + 0.834·23-s + 1/5·25-s − 1.11·29-s − 1.79·31-s + 1.64·37-s − 1.24·41-s − 0.304·43-s − 1.64·53-s + 0.809·55-s − 1.56·59-s + 1.28·61-s + 0.124·65-s + 1.63·73-s + 1.31·83-s + 0.410·95-s − 0.609·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 229320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(229320\)    =    \(2^{3} \cdot 3^{2} \cdot 5 \cdot 7^{2} \cdot 13\)
Sign: $-1$
Analytic conductor: \(1831.12\)
Root analytic conductor: \(42.7916\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{229320} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 229320,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + T \)
7 \( 1 \)
13 \( 1 + T \)
good11 \( 1 + 6 T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 10 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 14 T + p T^{2} \)
79 \( 1 + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 6 T + p T^{2} \)
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

−13.13738977287588, −12.64340174074170, −12.54449018525719, −11.65901698932897, −11.19762725149161, −10.81422691331529, −10.62529395246895, −9.760751641811977, −9.593681419929131, −8.858013665509564, −8.440805679995658, −7.880893834116752, −7.523911902671894, −7.201502977760560, −6.422531540131999, −6.012770693811492, −5.289875573224342, −4.977530930904770, −4.563421113764005, −3.691835219128706, −3.396718966293524, −2.663126497398616, −2.188942775268034, −1.599282379447885, −0.5454996395590500, 0, 0.5454996395590500, 1.599282379447885, 2.188942775268034, 2.663126497398616, 3.396718966293524, 3.691835219128706, 4.563421113764005, 4.977530930904770, 5.289875573224342, 6.012770693811492, 6.422531540131999, 7.201502977760560, 7.523911902671894, 7.880893834116752, 8.440805679995658, 8.858013665509564, 9.593681419929131, 9.760751641811977, 10.62529395246895, 10.81422691331529, 11.19762725149161, 11.65901698932897, 12.54449018525719, 12.64340174074170, 13.13738977287588

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