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Properties

Label 2-420e2-1.1-c1-0-259
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

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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.af
23 \( 1 + 6 T + p T^{2} \) 1.23.g
29 \( 1 - 4 T + p T^{2} \) 1.29.ae
31 \( 1 - 8 T + p T^{2} \) 1.31.ai
37 \( 1 + T + p T^{2} \) 1.37.b
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 - 8 T + p T^{2} \) 1.47.ai
53 \( 1 - 8 T + p T^{2} \) 1.53.ai
59 \( 1 + p T^{2} \) 1.59.a
61 \( 1 + 5 T + p T^{2} \) 1.61.f
67 \( 1 + 5 T + p T^{2} \) 1.67.f
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.m
97 \( 1 + 11 T + p T^{2} \) 1.97.l
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−13.57839319354847, −12.85438981117386, −12.30729813048248, −12.13228973987872, −11.74979907741798, −10.98505367897430, −10.35265050791990, −10.12511136420293, −9.799233060594554, −9.299672789152756, −8.478606595183099, −7.972740551200085, −7.732537853672091, −7.239802033450648, −6.804580111887278, −5.867212732758621, −5.486520170072100, −5.221352812925778, −4.471467859337450, −4.152253903893798, −3.102640140016508, −2.662489095362699, −2.506628152179176, −1.520890677058106, −0.6912711164795178, 0, 0.6912711164795178, 1.520890677058106, 2.506628152179176, 2.662489095362699, 3.102640140016508, 4.152253903893798, 4.471467859337450, 5.221352812925778, 5.486520170072100, 5.867212732758621, 6.804580111887278, 7.239802033450648, 7.732537853672091, 7.972740551200085, 8.478606595183099, 9.299672789152756, 9.799233060594554, 10.12511136420293, 10.35265050791990, 10.98505367897430, 11.74979907741798, 12.13228973987872, 12.30729813048248, 12.85438981117386, 13.57839319354847

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