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Properties

Label 2-609e2-1.1-c1-0-9
Degree $2$
Conductor $370881$
Sign $1$
Analytic cond. $2961.49$
Root an. cond. $54.4196$
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  + 2·2-s + 2·4-s + 2·5-s + 4·10-s − 2·11-s + 13-s − 4·16-s − 4·17-s + 4·19-s + 4·20-s − 4·22-s + 6·23-s − 25-s + 2·26-s − 7·31-s − 8·32-s − 8·34-s − 2·37-s + 8·38-s − 9·43-s − 4·44-s + 12·46-s + 10·47-s − 2·50-s + 2·52-s − 4·55-s − 6·59-s + ⋯
L(s)  = 1  + 1.41·2-s + 4-s + 0.894·5-s + 1.26·10-s − 0.603·11-s + 0.277·13-s − 16-s − 0.970·17-s + 0.917·19-s + 0.894·20-s − 0.852·22-s + 1.25·23-s − 1/5·25-s + 0.392·26-s − 1.25·31-s − 1.41·32-s − 1.37·34-s − 0.328·37-s + 1.29·38-s − 1.37·43-s − 0.603·44-s + 1.76·46-s + 1.45·47-s − 0.282·50-s + 0.277·52-s − 0.539·55-s − 0.781·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 370881 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(370881\)    =    \(3^{2} \cdot 7^{2} \cdot 29^{2}\)
Sign: $1$
Analytic conductor: \(2961.49\)
Root analytic conductor: \(54.4196\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 370881,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(5.052104230\)
\(L(\frac12)\) \(\approx\) \(5.052104230\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
29 \( 1 \)
good2 \( 1 - p T + p T^{2} \)
5 \( 1 - 2 T + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 - 6 T + p T^{2} \)
31 \( 1 + 7 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + p T^{2} \)
43 \( 1 + 9 T + p T^{2} \)
47 \( 1 - 10 T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 6 T + p T^{2} \)
61 \( 1 - 13 T + p T^{2} \)
67 \( 1 + 9 T + p T^{2} \)
71 \( 1 - 4 T + p T^{2} \)
73 \( 1 - 7 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 - 2 T + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−12.71628937626215, −12.22110817868767, −11.58683921158878, −11.32556440003548, −10.81696326392459, −10.38334249562227, −9.782357290818582, −9.351583134877072, −8.840680440094879, −8.596408899202685, −7.683639970572380, −7.303163398614281, −6.729690219288936, −6.414647332500047, −5.749216577758232, −5.435020329736064, −5.090641862887568, −4.612363655394039, −3.954487207996184, −3.496245665384413, −2.973045557092629, −2.443343562120371, −1.980804130511210, −1.340243435209276, −0.4212179600938924, 0.4212179600938924, 1.340243435209276, 1.980804130511210, 2.443343562120371, 2.973045557092629, 3.496245665384413, 3.954487207996184, 4.612363655394039, 5.090641862887568, 5.435020329736064, 5.749216577758232, 6.414647332500047, 6.729690219288936, 7.303163398614281, 7.683639970572380, 8.596408899202685, 8.840680440094879, 9.351583134877072, 9.782357290818582, 10.38334249562227, 10.81696326392459, 11.32556440003548, 11.58683921158878, 12.22110817868767, 12.71628937626215

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