↑

Properties

Label 2-420e2-1.1-c1-0-120
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 + 2·13-s − 3·17-s − 3·23-s + 6·29-s − 9·31-s − 5·41-s − 6·43-s − 9·47-s − 6·53-s + 8·59-s + 8·61-s + 14·67-s + 11·71-s + 2·73-s − 9·79-s + 6·83-s − 11·89-s + 11·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s − 0.727·17-s − 0.625·23-s + 1.11·29-s − 1.61·31-s − 0.780·41-s − 0.914·43-s − 1.31·47-s − 0.824·53-s + 1.04·59-s + 1.02·61-s + 1.71·67-s + 1.30·71-s + 0.234·73-s − 1.01·79-s + 0.658·83-s − 1.16·89-s + 1.11·97-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

\(L(1)\) \(\approx\) \(2.273530518\)
\(L(\frac12)\) \(\approx\) \(2.273530518\)
\(L(\frac{3}{2})\) 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 \( 1 - 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 3 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 9 T + p T^{2} \)
37 \( 1 + p T^{2} \)
41 \( 1 + 5 T + p T^{2} \)
43 \( 1 + 6 T + p T^{2} \)
47 \( 1 + 9 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 - 8 T + p T^{2} \)
67 \( 1 - 14 T + p T^{2} \)
71 \( 1 - 11 T + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 9 T + p T^{2} \)
83 \( 1 - 6 T + p T^{2} \)
89 \( 1 + 11 T + p T^{2} \)
97 \( 1 - 11 T + p T^{2} \)
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

−13.08559953612010, −12.82522618260893, −12.16376662600194, −11.76700179744004, −11.19369771255259, −11.07107235202608, −10.28747584751546, −9.757120054359243, −9.493907610170955, −8.792556570027071, −8.366439426852008, −8.160546039058849, −7.138662590952414, −6.936021589995061, −6.354393961023044, −6.015964528317053, −5.225354997457756, −4.816737981244142, −4.152644254295394, −3.582143979512623, −3.352590819068613, −2.315230953048633, −1.865800132911760, −1.235088937390446, −0.4469928496840127, 0.4469928496840127, 1.235088937390446, 1.865800132911760, 2.315230953048633, 3.352590819068613, 3.582143979512623, 4.152644254295394, 4.816737981244142, 5.225354997457756, 6.015964528317053, 6.354393961023044, 6.936021589995061, 7.138662590952414, 8.160546039058849, 8.366439426852008, 8.792556570027071, 9.493907610170955, 9.757120054359243, 10.28747584751546, 11.07107235202608, 11.19369771255259, 11.76700179744004, 12.16376662600194, 12.82522618260893, 13.08559953612010

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