Properties

Label 2-25200-1.1-c1-0-133
Degree $2$
Conductor $25200$
Sign $-1$
Analytic cond. $201.223$
Root an. cond. $14.1853$
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  + 7-s + 4·13-s − 6·17-s − 2·19-s + 6·23-s − 2·31-s − 2·37-s + 6·41-s − 4·43-s + 49-s + 6·53-s − 12·59-s − 10·61-s − 4·67-s − 12·71-s + 4·73-s − 8·79-s + 12·83-s + 6·89-s + 4·91-s − 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 6·119-s + ⋯
L(s)  = 1  + 0.377·7-s + 1.10·13-s − 1.45·17-s − 0.458·19-s + 1.25·23-s − 0.359·31-s − 0.328·37-s + 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.824·53-s − 1.56·59-s − 1.28·61-s − 0.488·67-s − 1.42·71-s + 0.468·73-s − 0.900·79-s + 1.31·83-s + 0.635·89-s + 0.419·91-s − 0.812·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.550·119-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 25200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(25200\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(201.223\)
Root analytic conductor: \(14.1853\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 25200,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 - T \)
good11 \( 1 + p T^{2} \)
13 \( 1 - 4 T + p T^{2} \)
17 \( 1 + 6 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 + 2 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 + 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 - 4 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 - 6 T + p T^{2} \)
97 \( 1 + 8 T + p T^{2} \)
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

−15.52133137573627, −15.13311207233201, −14.69122567877109, −13.85685043717855, −13.52493288948067, −13.01178805424512, −12.49775653166748, −11.72772260037185, −11.23891566502524, −10.72824707976540, −10.46956066257807, −9.413821565808620, −8.910208858842106, −8.664373025626433, −7.854942903905092, −7.274336451104369, −6.576984913874407, −6.148989826005222, −5.404192759346571, −4.655125136872928, −4.218524030660086, −3.408317167102460, −2.688179829594974, −1.851245820248783, −1.142383044673906, 0, 1.142383044673906, 1.851245820248783, 2.688179829594974, 3.408317167102460, 4.218524030660086, 4.655125136872928, 5.404192759346571, 6.148989826005222, 6.576984913874407, 7.274336451104369, 7.854942903905092, 8.664373025626433, 8.910208858842106, 9.413821565808620, 10.46956066257807, 10.72824707976540, 11.23891566502524, 11.72772260037185, 12.49775653166748, 13.01178805424512, 13.52493288948067, 13.85685043717855, 14.69122567877109, 15.13311207233201, 15.52133137573627

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