Properties

Label 2-5775-1.1-c1-0-167
Degree $2$
Conductor $5775$
Sign $-1$
Analytic cond. $46.1136$
Root an. cond. $6.79070$
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  − 2·2-s + 3-s + 2·4-s − 2·6-s + 7-s + 9-s + 11-s + 2·12-s + 3·13-s − 2·14-s − 4·16-s − 17-s − 2·18-s + 7·19-s + 21-s − 2·22-s − 4·23-s − 6·26-s + 27-s + 2·28-s − 8·29-s − 2·31-s + 8·32-s + 33-s + 2·34-s + 2·36-s − 11·37-s + ⋯
L(s)  = 1  − 1.41·2-s + 0.577·3-s + 4-s − 0.816·6-s + 0.377·7-s + 1/3·9-s + 0.301·11-s + 0.577·12-s + 0.832·13-s − 0.534·14-s − 16-s − 0.242·17-s − 0.471·18-s + 1.60·19-s + 0.218·21-s − 0.426·22-s − 0.834·23-s − 1.17·26-s + 0.192·27-s + 0.377·28-s − 1.48·29-s − 0.359·31-s + 1.41·32-s + 0.174·33-s + 0.342·34-s + 1/3·36-s − 1.80·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(5775\)    =    \(3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $-1$
Analytic conductor: \(46.1136\)
Root analytic conductor: \(6.79070\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 5775,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
good2 \( 1 + p T + p T^{2} \)
13 \( 1 - 3 T + p T^{2} \)
17 \( 1 + T + p T^{2} \)
19 \( 1 - 7 T + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 8 T + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 11 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 + 5 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - T + p T^{2} \)
67 \( 1 + 11 T + p T^{2} \)
71 \( 1 + 9 T + p T^{2} \)
73 \( 1 + 11 T + p T^{2} \)
79 \( 1 - 2 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 - 12 T + p T^{2} \)
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

−7.965803761309494142370444838378, −7.29711028435675921307862287590, −6.76299154488479888787527970953, −5.70936687271058520986134308885, −4.87734528671430316251373134870, −3.83339689022956820136420079858, −3.15687395648363653738084615950, −1.78585648821215823952733522406, −1.47289631681437433746547694179, 0, 1.47289631681437433746547694179, 1.78585648821215823952733522406, 3.15687395648363653738084615950, 3.83339689022956820136420079858, 4.87734528671430316251373134870, 5.70936687271058520986134308885, 6.76299154488479888787527970953, 7.29711028435675921307862287590, 7.965803761309494142370444838378

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