↑

Properties

Label 2-420e2-1.1-c1-0-208
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 $1$

Origins

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 6·11-s − 7·13-s + 4·17-s − 5·19-s + 6·23-s + 4·29-s − 8·31-s + 37-s − 2·41-s − 4·43-s + 8·47-s − 8·53-s + 5·61-s + 5·67-s + 4·71-s + 73-s − 11·79-s − 16·83-s + 12·89-s − 11·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.80·11-s − 1.94·13-s + 0.970·17-s − 1.14·19-s + 1.25·23-s + 0.742·29-s − 1.43·31-s + 0.164·37-s − 0.312·41-s − 0.609·43-s + 1.16·47-s − 1.09·53-s + 0.640·61-s + 0.610·67-s + 0.474·71-s + 0.117·73-s − 1.23·79-s − 1.75·83-s + 1.27·89-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: \(1\)
Selberg data: \((2,\ 176400,\ (\ :1/2),\ -1)\)

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)$Isogeny Class over $\mathbf{F}_p$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 + 6 T + p T^{2} \) 1.11.g
13 \( 1 + 7 T + p T^{2} \) 1.13.h
17 \( 1 - 4 T + p T^{2} \) 1.17.ae
19 \( 1 + 5 T + p T^{2} \) 1.19.f
23 \( 1 - 6 T + p T^{2} \) 1.23.ag
29 \( 1 - 4 T + p T^{2} \) 1.29.ae
31 \( 1 + 8 T + p T^{2} \) 1.31.i
37 \( 1 - T + p T^{2} \) 1.37.ab
41 \( 1 + 2 T + p T^{2} \) 1.41.c
43 \( 1 + 4 T + p T^{2} \) 1.43.e
47 \( 1 - 8 T + p T^{2} \) 1.47.ai
53 \( 1 + 8 T + p T^{2} \) 1.53.i
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 - 5 T + p T^{2} \) 1.61.af
67 \( 1 - 5 T + p T^{2} \) 1.67.af
71 \( 1 - 4 T + p T^{2} \) 1.71.ae
73 \( 1 - T + p T^{2} \) 1.73.ab
79 \( 1 + 11 T + p T^{2} \) 1.79.l
83 \( 1 + 16 T + p T^{2} \) 1.83.q
89 \( 1 - 12 T + p T^{2} \) 1.89.am
97 \( 1 + 11 T + p T^{2} \) 1.97.l
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

−13.20596011837040, −12.90158945884640, −12.50621003157493, −12.23406806859128, −11.48480256456819, −10.98610426568315, −10.47716887208142, −10.22269238450560, −9.630447974303475, −9.292171714669152, −8.411732206196483, −8.244108781769280, −7.530222060267304, −7.223823607010502, −6.834165891043539, −5.982216474187259, −5.404938122685545, −5.107885993369040, −4.675192869973766, −4.018009440436737, −3.150566890038706, −2.775444528309910, −2.302068653306595, −1.648814342830649, −0.6065096037100650, 0, 0.6065096037100650, 1.648814342830649, 2.302068653306595, 2.775444528309910, 3.150566890038706, 4.018009440436737, 4.675192869973766, 5.107885993369040, 5.404938122685545, 5.982216474187259, 6.834165891043539, 7.223823607010502, 7.530222060267304, 8.244108781769280, 8.411732206196483, 9.292171714669152, 9.630447974303475, 10.22269238450560, 10.47716887208142, 10.98610426568315, 11.48480256456819, 12.23406806859128, 12.50621003157493, 12.90158945884640, 13.20596011837040

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