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Properties

Label 2-420e2-1.1-c1-0-408
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 − 13-s − 4·17-s + 19-s + 4·23-s − 4·29-s − 5·31-s + 6·37-s + 12·41-s + 5·43-s + 8·47-s − 12·53-s − 8·59-s − 7·61-s + 13·67-s + 12·71-s − 6·73-s − 12·79-s − 8·83-s − 13·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s − 0.277·13-s − 0.970·17-s + 0.229·19-s + 0.834·23-s − 0.742·29-s − 0.898·31-s + 0.986·37-s + 1.87·41-s + 0.762·43-s + 1.16·47-s − 1.64·53-s − 1.04·59-s − 0.896·61-s + 1.58·67-s + 1.42·71-s − 0.702·73-s − 1.35·79-s − 0.878·83-s − 1.31·97-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 \( 1 + T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 + 4 T + p T^{2} \)
31 \( 1 + 5 T + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 12 T + p T^{2} \)
43 \( 1 - 5 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 + 8 T + p T^{2} \)
61 \( 1 + 7 T + p T^{2} \)
67 \( 1 - 13 T + p T^{2} \)
71 \( 1 - 12 T + p T^{2} \)
73 \( 1 + 6 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 8 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 + 13 T + p T^{2} \)
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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

−13.36375440975426, −12.87694935329894, −12.50023159937786, −12.11969979074481, −11.29022487794138, −11.08935949414481, −10.89639439103946, −9.987070429788540, −9.375043647860118, −9.266635313866905, −8.837940833408086, −8.106345707414325, −7.544208668536036, −7.184450020976092, −6.626644401831457, −6.106628690637605, −5.680787126344393, −5.019716808196639, −4.278058598300021, −4.168236129523044, −3.396006522279147, −2.747481616717477, −2.204369418784622, −1.462997786077040, −0.8956898902284296, 0, 0.8956898902284296, 1.462997786077040, 2.204369418784622, 2.747481616717477, 3.396006522279147, 4.168236129523044, 4.278058598300021, 5.019716808196639, 5.680787126344393, 6.106628690637605, 6.626644401831457, 7.184450020976092, 7.544208668536036, 8.106345707414325, 8.837940833408086, 9.266635313866905, 9.375043647860118, 9.987070429788540, 10.89639439103946, 11.08935949414481, 11.29022487794138, 12.11969979074481, 12.50023159937786, 12.87694935329894, 13.36375440975426

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