Properties

Label 2-25350-1.1-c1-0-83
Degree $2$
Conductor $25350$
Sign $-1$
Analytic cond. $202.420$
Root an. cond. $14.2274$
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-s + 4-s + 6-s + 2·7-s + 8-s + 9-s − 4·11-s + 12-s + 2·14-s + 16-s − 8·17-s + 18-s + 6·19-s + 2·21-s − 4·22-s − 6·23-s + 24-s + 27-s + 2·28-s − 4·29-s + 32-s − 4·33-s − 8·34-s + 36-s − 2·37-s + 6·38-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 1/2·4-s + 0.408·6-s + 0.755·7-s + 0.353·8-s + 1/3·9-s − 1.20·11-s + 0.288·12-s + 0.534·14-s + 1/4·16-s − 1.94·17-s + 0.235·18-s + 1.37·19-s + 0.436·21-s − 0.852·22-s − 1.25·23-s + 0.204·24-s + 0.192·27-s + 0.377·28-s − 0.742·29-s + 0.176·32-s − 0.696·33-s − 1.37·34-s + 1/6·36-s − 0.328·37-s + 0.973·38-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25350 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25350\)    =    \(2 \cdot 3 \cdot 5^{2} \cdot 13^{2}\)
Sign: $-1$
Analytic conductor: \(202.420\)
Root analytic conductor: \(14.2274\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{25350} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25350,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 - T \)
5 \( 1 \)
13 \( 1 \)
good7 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 4 T + p T^{2} \)
17 \( 1 + 8 T + p T^{2} \)
19 \( 1 - 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 10 T + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 - 12 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 + 8 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 14 T + p T^{2} \)
97 \( 1 + 16 T + p T^{2} \)
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

−15.48996666109446, −15.09618859696784, −14.51692821800000, −13.89630803618822, −13.46659216808787, −13.26073572339376, −12.38554408699422, −11.99390298909900, −11.22596647057876, −10.87473701002950, −10.33219501379823, −9.512783808667228, −9.087612649357615, −8.180004144801207, −7.922145861718424, −7.311305268683853, −6.682696217974153, −5.912328329083025, −5.236396327264628, −4.802323356236406, −4.076516479434611, −3.520474223899625, −2.476713354488038, −2.294393118311868, −1.347876289821976, 0, 1.347876289821976, 2.294393118311868, 2.476713354488038, 3.520474223899625, 4.076516479434611, 4.802323356236406, 5.236396327264628, 5.912328329083025, 6.682696217974153, 7.311305268683853, 7.922145861718424, 8.180004144801207, 9.087612649357615, 9.512783808667228, 10.33219501379823, 10.87473701002950, 11.22596647057876, 11.99390298909900, 12.38554408699422, 13.26073572339376, 13.46659216808787, 13.89630803618822, 14.51692821800000, 15.09618859696784, 15.48996666109446

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