↑

Properties

Label 2-28322-1.1-c1-0-18
Degree $2$
Conductor $28322$
Sign $-1$
Analytic cond. $226.152$
Root an. cond. $15.0383$
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  + 2-s − 3·3-s + 4-s − 3·6-s + 8-s + 6·9-s + 3·11-s − 3·12-s − 5·13-s + 16-s + 6·18-s + 4·19-s + 3·22-s + 4·23-s − 3·24-s − 5·25-s − 5·26-s − 9·27-s + 4·29-s − 4·31-s + 32-s − 9·33-s + 6·36-s + 8·37-s + 4·38-s + 15·39-s − 4·41-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.73·3-s + 1/2·4-s − 1.22·6-s + 0.353·8-s + 2·9-s + 0.904·11-s − 0.866·12-s − 1.38·13-s + 1/4·16-s + 1.41·18-s + 0.917·19-s + 0.639·22-s + 0.834·23-s − 0.612·24-s − 25-s − 0.980·26-s − 1.73·27-s + 0.742·29-s − 0.718·31-s + 0.176·32-s − 1.56·33-s + 36-s + 1.31·37-s + 0.648·38-s + 2.40·39-s − 0.624·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 28322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(28322\)    =    \(2 \cdot 7^{2} \cdot 17^{2}\)
Sign: $-1$
Analytic conductor: \(226.152\)
Root analytic conductor: \(15.0383\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 28322,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
7 \( 1 \)
17 \( 1 \)
good3 \( 1 + p T + p T^{2} \)
5 \( 1 + p T^{2} \)
11 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 4 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 - 8 T + p T^{2} \)
41 \( 1 + 4 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 + 16 T + p T^{2} \)
71 \( 1 + T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 - 7 T + p T^{2} \)
83 \( 1 + 16 T + p T^{2} \)
89 \( 1 + 9 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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

−15.56455972150446, −14.78282032628614, −14.65386121707296, −13.74788820842312, −13.23921793124176, −12.67502693113326, −12.10526299420910, −11.76905890191424, −11.44360553398967, −10.88356254008985, −10.04770999483931, −9.861151099153422, −9.171546257665538, −8.161060113063965, −7.381907290903440, −6.983230590068153, −6.506710072739805, −5.833116880582711, −5.335391284521367, −4.831254762587714, −4.332411349054978, −3.578131000333179, −2.716555499718257, −1.723877462446315, −1.000145090276563, 0, 1.000145090276563, 1.723877462446315, 2.716555499718257, 3.578131000333179, 4.332411349054978, 4.831254762587714, 5.335391284521367, 5.833116880582711, 6.506710072739805, 6.983230590068153, 7.381907290903440, 8.161060113063965, 9.171546257665538, 9.861151099153422, 10.04770999483931, 10.88356254008985, 11.44360553398967, 11.76905890191424, 12.10526299420910, 12.67502693113326, 13.23921793124176, 13.74788820842312, 14.65386121707296, 14.78282032628614, 15.56455972150446

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