Properties

Label 2-2450-1.1-c3-0-191
Degree $2$
Conductor $2450$
Sign $-1$
Analytic cond. $144.554$
Root an. cond. $12.0230$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2·2-s + 7.07·3-s + 4·4-s + 14.1·6-s + 8·8-s + 23.0·9-s − 14·11-s + 28.2·12-s − 50.9·13-s + 16·16-s − 1.41·17-s + 46.0·18-s − 1.41·19-s − 28·22-s − 140·23-s + 56.5·24-s − 101.·26-s − 28.2·27-s − 286·29-s − 93.3·31-s + 32·32-s − 98.9·33-s − 2.82·34-s + 92.0·36-s + 38·37-s − 2.82·38-s − 360·39-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.36·3-s + 0.5·4-s + 0.962·6-s + 0.353·8-s + 0.851·9-s − 0.383·11-s + 0.680·12-s − 1.08·13-s + 0.250·16-s − 0.0201·17-s + 0.602·18-s − 0.0170·19-s − 0.271·22-s − 1.26·23-s + 0.481·24-s − 0.768·26-s − 0.201·27-s − 1.83·29-s − 0.540·31-s + 0.176·32-s − 0.522·33-s − 0.0142·34-s + 0.425·36-s + 0.168·37-s − 0.0120·38-s − 1.47·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2450 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2450\)    =    \(2 \cdot 5^{2} \cdot 7^{2}\)
Sign: $-1$
Analytic conductor: \(144.554\)
Root analytic conductor: \(12.0230\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{2450} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2450,\ (\ :3/2),\ -1)\)

Particular Values

\(L(2)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2T \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 - 7.07T + 27T^{2} \)
11 \( 1 + 14T + 1.33e3T^{2} \)
13 \( 1 + 50.9T + 2.19e3T^{2} \)
17 \( 1 + 1.41T + 4.91e3T^{2} \)
19 \( 1 + 1.41T + 6.85e3T^{2} \)
23 \( 1 + 140T + 1.21e4T^{2} \)
29 \( 1 + 286T + 2.43e4T^{2} \)
31 \( 1 + 93.3T + 2.97e4T^{2} \)
37 \( 1 - 38T + 5.06e4T^{2} \)
41 \( 1 + 125.T + 6.89e4T^{2} \)
43 \( 1 - 34T + 7.95e4T^{2} \)
47 \( 1 + 523.T + 1.03e5T^{2} \)
53 \( 1 - 74T + 1.48e5T^{2} \)
59 \( 1 - 434.T + 2.05e5T^{2} \)
61 \( 1 - 14.1T + 2.26e5T^{2} \)
67 \( 1 + 684T + 3.00e5T^{2} \)
71 \( 1 - 588T + 3.57e5T^{2} \)
73 \( 1 - 270.T + 3.89e5T^{2} \)
79 \( 1 - 1.22e3T + 4.93e5T^{2} \)
83 \( 1 + 422.T + 5.71e5T^{2} \)
89 \( 1 - 618.T + 7.04e5T^{2} \)
97 \( 1 + 1.48e3T + 9.12e5T^{2} \)
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

−7.960028318862136626563638551732, −7.65920894953528450080644322598, −6.77841049375781973582204899012, −5.73092852419456904223337723912, −4.95061646402822600418373974149, −3.95307823322392883437638075759, −3.34081760760614279224756785572, −2.36916403817894882421215005140, −1.84637492650488867793094513198, 0, 1.84637492650488867793094513198, 2.36916403817894882421215005140, 3.34081760760614279224756785572, 3.95307823322392883437638075759, 4.95061646402822600418373974149, 5.73092852419456904223337723912, 6.77841049375781973582204899012, 7.65920894953528450080644322598, 7.960028318862136626563638551732

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