Properties

Label 2-273600-1.1-c1-0-357
Degree $2$
Conductor $273600$
Sign $-1$
Analytic cond. $2184.70$
Root an. cond. $46.7408$
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 + 2·13-s + 2·17-s + 19-s + 4·23-s + 6·29-s + 4·31-s − 6·37-s − 10·41-s − 4·43-s − 12·47-s − 7·49-s − 6·53-s − 12·59-s + 2·61-s + 4·67-s − 8·71-s + 6·73-s − 4·79-s + 12·83-s − 10·89-s − 2·97-s + 101-s + 103-s + 107-s + 109-s + 113-s + ⋯
L(s)  = 1  + 1.20·11-s + 0.554·13-s + 0.485·17-s + 0.229·19-s + 0.834·23-s + 1.11·29-s + 0.718·31-s − 0.986·37-s − 1.56·41-s − 0.609·43-s − 1.75·47-s − 49-s − 0.824·53-s − 1.56·59-s + 0.256·61-s + 0.488·67-s − 0.949·71-s + 0.702·73-s − 0.450·79-s + 1.31·83-s − 1.05·89-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(273600\)    =    \(2^{6} \cdot 3^{2} \cdot 5^{2} \cdot 19\)
Sign: $-1$
Analytic conductor: \(2184.70\)
Root analytic conductor: \(46.7408\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{273600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 273600,\ (\ :1/2),\ -1)\)

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 \( 1 \)
19 \( 1 - T \)
good7 \( 1 + p T^{2} \)
11 \( 1 - 4 T + p T^{2} \)
13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
23 \( 1 - 4 T + p T^{2} \)
29 \( 1 - 6 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 10 T + p T^{2} \)
43 \( 1 + 4 T + p T^{2} \)
47 \( 1 + 12 T + p T^{2} \)
53 \( 1 + 6 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 2 T + p T^{2} \)
67 \( 1 - 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + 10 T + p T^{2} \)
97 \( 1 + 2 T + p T^{2} \)
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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

−13.12411026087384, −12.34862226737305, −12.13962100860875, −11.58710489443088, −11.30616219889140, −10.68466727121458, −10.23298164098836, −9.710470104651949, −9.393074319905420, −8.739712950924802, −8.392840205427743, −8.014627296484275, −7.313859623469781, −6.692652255733004, −6.524136343498661, −6.054440867992387, −5.256959005322891, −4.852865914643999, −4.449439808933340, −3.616496137839539, −3.289884005766156, −2.906236589393143, −1.830183796887723, −1.487263003795853, −0.9130594715952859, 0, 0.9130594715952859, 1.487263003795853, 1.830183796887723, 2.906236589393143, 3.289884005766156, 3.616496137839539, 4.449439808933340, 4.852865914643999, 5.256959005322891, 6.054440867992387, 6.524136343498661, 6.692652255733004, 7.313859623469781, 8.014627296484275, 8.392840205427743, 8.739712950924802, 9.393074319905420, 9.710470104651949, 10.23298164098836, 10.68466727121458, 11.30616219889140, 11.58710489443088, 12.13962100860875, 12.34862226737305, 13.12411026087384

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