↑

Properties

Label 2-420e2-1.1-c1-0-103
Degree $2$
Conductor $176400$
Sign $1$
Analytic cond. $1408.56$
Root an. cond. $37.5308$
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  + 4·11-s − 8·23-s − 2·29-s + 6·37-s − 12·43-s − 10·53-s + 4·67-s + 16·71-s − 8·79-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 1.66·23-s − 0.371·29-s + 0.986·37-s − 1.82·43-s − 1.37·53-s + 0.488·67-s + 1.89·71-s − 0.900·79-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 176400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(176400\)    =    \(2^{4} \cdot 3^{2} \cdot 5^{2} \cdot 7^{2}\)
Sign: $1$
Analytic conductor: \(1408.56\)
Root analytic conductor: \(37.5308\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.996715743\)
\(L(\frac12)\) \(\approx\) \(1.996715743\)
\(L(\frac{3}{2})\) 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 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + p T^{2} \)
17 \( 1 + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 8 T + p T^{2} \)
29 \( 1 + 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 12 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 + 10 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 - 16 T + p T^{2} \)
73 \( 1 + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + p T^{2} \)
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

−13.10992057925595, −12.73025744562693, −12.14768339063632, −11.81864500084131, −11.26348428260910, −11.03516602115974, −10.17633773307906, −9.770610899889874, −9.577756448379475, −8.802636928879287, −8.447545145637491, −7.912858197619231, −7.422674882677396, −6.812913183368841, −6.247733419990202, −6.077492099334793, −5.305193822013215, −4.701907644440958, −4.225522293516936, −3.609741805093243, −3.289678972343360, −2.338213619461523, −1.857689294944980, −1.250027034767230, −0.4140728912804747, 0.4140728912804747, 1.250027034767230, 1.857689294944980, 2.338213619461523, 3.289678972343360, 3.609741805093243, 4.225522293516936, 4.701907644440958, 5.305193822013215, 6.077492099334793, 6.247733419990202, 6.812913183368841, 7.422674882677396, 7.912858197619231, 8.447545145637491, 8.802636928879287, 9.577756448379475, 9.770610899889874, 10.17633773307906, 11.03516602115974, 11.26348428260910, 11.81864500084131, 12.14768339063632, 12.73025744562693, 13.10992057925595

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