Properties

Label 2-369600-1.1-c1-0-104
Degree $2$
Conductor $369600$
Sign $1$
Analytic cond. $2951.27$
Root an. cond. $54.3256$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 7-s + 9-s − 11-s + 2·13-s + 2·17-s + 21-s − 4·23-s + 27-s − 6·29-s − 4·31-s − 33-s − 10·37-s + 2·39-s − 2·41-s + 4·43-s + 8·47-s + 49-s + 2·51-s + 2·53-s + 4·59-s + 10·61-s + 63-s − 4·67-s − 4·69-s − 8·71-s + 10·73-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.377·7-s + 1/3·9-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 0.218·21-s − 0.834·23-s + 0.192·27-s − 1.11·29-s − 0.718·31-s − 0.174·33-s − 1.64·37-s + 0.320·39-s − 0.312·41-s + 0.609·43-s + 1.16·47-s + 1/7·49-s + 0.280·51-s + 0.274·53-s + 0.520·59-s + 1.28·61-s + 0.125·63-s − 0.488·67-s − 0.481·69-s − 0.949·71-s + 1.17·73-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(369600\)    =    \(2^{6} \cdot 3 \cdot 5^{2} \cdot 7 \cdot 11\)
Sign: $1$
Analytic conductor: \(2951.27\)
Root analytic conductor: \(54.3256\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{369600} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 369600,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.637690724\)
\(L(\frac12)\) \(\approx\) \(2.637690724\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
5 \( 1 \)
7 \( 1 - T \)
11 \( 1 + T \)
good13 \( 1 - 2 T + p T^{2} \)
17 \( 1 - 2 T + p T^{2} \)
19 \( 1 + p T^{2} \)
23 \( 1 + 4 T + p T^{2} \)
29 \( 1 + 6 T + p T^{2} \)
31 \( 1 + 4 T + p T^{2} \)
37 \( 1 + 10 T + p T^{2} \)
41 \( 1 + 2 T + p T^{2} \)
43 \( 1 - 4 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 2 T + p T^{2} \)
59 \( 1 - 4 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 - 8 T + p T^{2} \)
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

−12.44018278084186, −12.13500735409567, −11.63027946987214, −11.04551383847462, −10.73328745462871, −10.20028098731304, −9.830389927791899, −9.241192456832830, −8.795460613272624, −8.484536097372055, −7.855269579646747, −7.608214693401848, −7.018721681059886, −6.628923541378298, −5.860320393687580, −5.436780681303918, −5.191763874889396, −4.258994979861210, −3.936207576618538, −3.529049601713302, −2.869229511110328, −2.269932190698050, −1.767604973507826, −1.249666720888109, −0.3954933939254357, 0.3954933939254357, 1.249666720888109, 1.767604973507826, 2.269932190698050, 2.869229511110328, 3.529049601713302, 3.936207576618538, 4.258994979861210, 5.191763874889396, 5.436780681303918, 5.860320393687580, 6.628923541378298, 7.018721681059886, 7.608214693401848, 7.855269579646747, 8.484536097372055, 8.795460613272624, 9.241192456832830, 9.830389927791899, 10.20028098731304, 10.73328745462871, 11.04551383847462, 11.63027946987214, 12.13500735409567, 12.44018278084186

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