Properties

Label 2-2925-1.1-c1-0-8
Degree $2$
Conductor $2925$
Sign $1$
Analytic cond. $23.3562$
Root an. cond. $4.83282$
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-s − 4-s + 3·8-s − 4·11-s − 13-s − 16-s + 2·17-s − 4·19-s + 4·22-s + 8·23-s + 26-s + 2·29-s − 8·31-s − 5·32-s − 2·34-s − 6·37-s + 4·38-s + 6·41-s + 4·43-s + 4·44-s − 8·46-s − 8·47-s − 7·49-s + 52-s + 6·53-s − 2·58-s + 12·59-s + ⋯
L(s)  = 1  − 0.707·2-s − 1/2·4-s + 1.06·8-s − 1.20·11-s − 0.277·13-s − 1/4·16-s + 0.485·17-s − 0.917·19-s + 0.852·22-s + 1.66·23-s + 0.196·26-s + 0.371·29-s − 1.43·31-s − 0.883·32-s − 0.342·34-s − 0.986·37-s + 0.648·38-s + 0.937·41-s + 0.609·43-s + 0.603·44-s − 1.17·46-s − 1.16·47-s − 49-s + 0.138·52-s + 0.824·53-s − 0.262·58-s + 1.56·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2925 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2925\)    =    \(3^{2} \cdot 5^{2} \cdot 13\)
Sign: $1$
Analytic conductor: \(23.3562\)
Root analytic conductor: \(4.83282\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2925,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7612303932\)
\(L(\frac12)\) \(\approx\) \(0.7612303932\)
\(L(\frac{3}{2})\) 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 \( 1 \)
13 \( 1 + T \)
good2 \( 1 + T + p T^{2} \)
7 \( 1 + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 - 8 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + 8 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + 8 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 12 T + p T^{2} \)
61 \( 1 + 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 - 16 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 18 T + p T^{2} \)
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

−8.728931928185248388257372058485, −8.136025378033966872311574564843, −7.43193712425490303306328835346, −6.75028913136328658806730701739, −5.44064201059248414897390242722, −5.04341051321205815078664291033, −4.05811586767484409743342375016, −3.01337399574887020794792151462, −1.90651535301238430158520971763, −0.59299171903779565408875923090, 0.59299171903779565408875923090, 1.90651535301238430158520971763, 3.01337399574887020794792151462, 4.05811586767484409743342375016, 5.04341051321205815078664291033, 5.44064201059248414897390242722, 6.75028913136328658806730701739, 7.43193712425490303306328835346, 8.136025378033966872311574564843, 8.728931928185248388257372058485

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