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Properties

Label 2-420e2-1.1-c1-0-480
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 − 3·13-s + 7·17-s − 6·19-s + 9·23-s − 3·29-s − 7·31-s + 10·37-s − 41-s + 13·43-s + 2·47-s + 53-s + 11·59-s − 13·61-s + 8·71-s + 8·73-s − 4·79-s − 7·83-s − 14·89-s + 8·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.832·13-s + 1.69·17-s − 1.37·19-s + 1.87·23-s − 0.557·29-s − 1.25·31-s + 1.64·37-s − 0.156·41-s + 1.98·43-s + 0.291·47-s + 0.137·53-s + 1.43·59-s − 1.66·61-s + 0.949·71-s + 0.936·73-s − 0.450·79-s − 0.768·83-s − 1.48·89-s + 0.812·97-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)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good11 \( 1 - 4 T + p T^{2} \)
13 \( 1 + 3 T + p T^{2} \)
17 \( 1 - 7 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 - 9 T + p T^{2} \)
29 \( 1 + 3 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 - 10 T + p T^{2} \)
41 \( 1 + T + p T^{2} \)
43 \( 1 - 13 T + p T^{2} \)
47 \( 1 - 2 T + p T^{2} \)
53 \( 1 - T + p T^{2} \)
59 \( 1 - 11 T + p T^{2} \)
61 \( 1 + 13 T + p T^{2} \)
67 \( 1 + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 8 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 + 7 T + p T^{2} \)
89 \( 1 + 14 T + p T^{2} \)
97 \( 1 - 8 T + p T^{2} \)
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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

−13.26917544391024, −12.78351106279859, −12.58138007362991, −12.10395342583136, −11.42261687657243, −11.16608739513298, −10.59326230062214, −10.12977455138523, −9.379397836755433, −9.278681735464714, −8.819906107862674, −8.005541024285601, −7.675227320887434, −7.081471530507281, −6.748747778546165, −6.039734895306151, −5.616365582149028, −5.093320368300469, −4.430034158375488, −3.938774970151967, −3.480493904187778, −2.684670318747983, −2.305732962311602, −1.305279650432800, −1.016489447241242, 0, 1.016489447241242, 1.305279650432800, 2.305732962311602, 2.684670318747983, 3.480493904187778, 3.938774970151967, 4.430034158375488, 5.093320368300469, 5.616365582149028, 6.039734895306151, 6.748747778546165, 7.081471530507281, 7.675227320887434, 8.005541024285601, 8.819906107862674, 9.278681735464714, 9.379397836755433, 10.12977455138523, 10.59326230062214, 11.16608739513298, 11.42261687657243, 12.10395342583136, 12.58138007362991, 12.78351106279859, 13.26917544391024

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