Properties

Label 2-2300-1.1-c1-0-7
Degree $2$
Conductor $2300$
Sign $1$
Analytic cond. $18.3655$
Root an. cond. $4.28550$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4·7-s − 2·9-s − 6·11-s + 13-s + 2·19-s − 4·21-s − 23-s + 5·27-s + 9·29-s + 5·31-s + 6·33-s − 2·37-s − 39-s − 9·41-s + 4·43-s + 3·47-s + 9·49-s + 6·53-s − 2·57-s + 2·61-s − 8·63-s + 10·67-s + 69-s − 3·71-s + 7·73-s − 24·77-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.51·7-s − 2/3·9-s − 1.80·11-s + 0.277·13-s + 0.458·19-s − 0.872·21-s − 0.208·23-s + 0.962·27-s + 1.67·29-s + 0.898·31-s + 1.04·33-s − 0.328·37-s − 0.160·39-s − 1.40·41-s + 0.609·43-s + 0.437·47-s + 9/7·49-s + 0.824·53-s − 0.264·57-s + 0.256·61-s − 1.00·63-s + 1.22·67-s + 0.120·69-s − 0.356·71-s + 0.819·73-s − 2.73·77-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2300 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2300\)    =    \(2^{2} \cdot 5^{2} \cdot 23\)
Sign: $1$
Analytic conductor: \(18.3655\)
Root analytic conductor: \(4.28550\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2300} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2300,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.416757087\)
\(L(\frac12)\) \(\approx\) \(1.416757087\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
23 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
7 \( 1 - 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 - 2 T + p T^{2} \)
29 \( 1 - 9 T + p T^{2} \)
31 \( 1 - 5 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 9 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 10 T + p T^{2} \)
71 \( 1 + 3 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 10 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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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

−8.658045597274035577262987857321, −8.263039158680734767784440545325, −7.65430632633120695676100199465, −6.63535022330794595242647107432, −5.59637642223260842167274984298, −5.14665972928625356317714150392, −4.50125749316712544034107584040, −3.05814912055831361324108561222, −2.18430524087619453430106080114, −0.790789825015232224154403917311, 0.790789825015232224154403917311, 2.18430524087619453430106080114, 3.05814912055831361324108561222, 4.50125749316712544034107584040, 5.14665972928625356317714150392, 5.59637642223260842167274984298, 6.63535022330794595242647107432, 7.65430632633120695676100199465, 8.263039158680734767784440545325, 8.658045597274035577262987857321

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