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Properties

Label 2-420e2-1.1-c1-0-433
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  − 4·11-s + 4·13-s + 2·17-s + 4·23-s + 8·31-s − 2·37-s + 2·41-s + 4·43-s − 4·47-s + 4·53-s + 4·59-s + 8·61-s − 4·67-s + 12·71-s − 4·73-s − 4·79-s + 8·83-s − 14·89-s − 12·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  − 1.20·11-s + 1.10·13-s + 0.485·17-s + 0.834·23-s + 1.43·31-s − 0.328·37-s + 0.312·41-s + 0.609·43-s − 0.583·47-s + 0.549·53-s + 0.520·59-s + 1.02·61-s − 0.488·67-s + 1.42·71-s − 0.468·73-s − 0.450·79-s + 0.878·83-s − 1.48·89-s − 1.21·97-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 \( 1 + 4 T + p T^{2} \) 1.11.e
13 \( 1 - 4 T + p T^{2} \) 1.13.ae
17 \( 1 - 2 T + p T^{2} \) 1.17.ac
19 \( 1 + p T^{2} \) 1.19.a
23 \( 1 - 4 T + p T^{2} \) 1.23.ae
29 \( 1 + p T^{2} \) 1.29.a
31 \( 1 - 8 T + p T^{2} \) 1.31.ai
37 \( 1 + 2 T + p T^{2} \) 1.37.c
41 \( 1 - 2 T + p T^{2} \) 1.41.ac
43 \( 1 - 4 T + p T^{2} \) 1.43.ae
47 \( 1 + 4 T + p T^{2} \) 1.47.e
53 \( 1 - 4 T + p T^{2} \) 1.53.ae
59 \( 1 - 4 T + p T^{2} \) 1.59.ae
61 \( 1 - 8 T + p T^{2} \) 1.61.ai
67 \( 1 + 4 T + p T^{2} \) 1.67.e
71 \( 1 - 12 T + p T^{2} \) 1.71.am
73 \( 1 + 4 T + p T^{2} \) 1.73.e
79 \( 1 + 4 T + p T^{2} \) 1.79.e
83 \( 1 - 8 T + p T^{2} \) 1.83.ai
89 \( 1 + 14 T + p T^{2} \) 1.89.o
97 \( 1 + 12 T + p T^{2} \) 1.97.m
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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

−13.42075072637982, −13.05406121755345, −12.45865174507328, −12.10463122393308, −11.41261136651445, −11.05633388324192, −10.59575022480282, −10.18051091563327, −9.646902126743875, −9.159075548542496, −8.462759855811676, −8.201188469948823, −7.794144940591474, −7.045512583038245, −6.695393827680742, −6.085502419019168, −5.373067448856930, −5.316293671880915, −4.451267701569911, −3.980324445509054, −3.314194701275634, −2.760157680288226, −2.338032840462533, −1.352900778792389, −0.9256439143422097, 0, 0.9256439143422097, 1.352900778792389, 2.338032840462533, 2.760157680288226, 3.314194701275634, 3.980324445509054, 4.451267701569911, 5.316293671880915, 5.373067448856930, 6.085502419019168, 6.695393827680742, 7.045512583038245, 7.794144940591474, 8.201188469948823, 8.462759855811676, 9.159075548542496, 9.646902126743875, 10.18051091563327, 10.59575022480282, 11.05633388324192, 11.41261136651445, 12.10463122393308, 12.45865174507328, 13.05406121755345, 13.42075072637982

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