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Properties

Label 2-609e2-1.1-c1-0-15
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 + 5-s − 2·10-s + 2·11-s + 7·13-s − 4·16-s + 4·17-s + 4·19-s + 2·20-s − 4·22-s − 4·25-s − 14·26-s + 8·31-s + 8·32-s − 8·34-s − 2·37-s − 8·38-s − 12·41-s + 6·43-s + 4·44-s + 8·47-s + 8·50-s + 14·52-s + 9·53-s + 2·55-s − 12·59-s + ⋯
L(s)  = 1  − 1.41·2-s + 4-s + 0.447·5-s − 0.632·10-s + 0.603·11-s + 1.94·13-s − 16-s + 0.970·17-s + 0.917·19-s + 0.447·20-s − 0.852·22-s − 4/5·25-s − 2.74·26-s + 1.43·31-s + 1.41·32-s − 1.37·34-s − 0.328·37-s − 1.29·38-s − 1.87·41-s + 0.914·43-s + 0.603·44-s + 1.16·47-s + 1.13·50-s + 1.94·52-s + 1.23·53-s + 0.269·55-s − 1.56·59-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

\(L(1)\) \(\approx\) \(2.031960500\)
\(L(\frac12)\) \(\approx\) \(2.031960500\)
\(L(\frac{3}{2})\) 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 \( 1 - T + p T^{2} \)
11 \( 1 - 2 T + p T^{2} \)
13 \( 1 - 7 T + p T^{2} \)
17 \( 1 - 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 + 2 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 8 T + p T^{2} \)
53 \( 1 - 9 T + p T^{2} \)
59 \( 1 + 12 T + p T^{2} \)
61 \( 1 - 10 T + p T^{2} \)
67 \( 1 + 12 T + p T^{2} \)
71 \( 1 - 5 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 12 T + p T^{2} \)
83 \( 1 + 11 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 18 T + p T^{2} \)
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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

−12.22080907911534, −11.83474575539879, −11.56513799239339, −11.04075227962598, −10.45431221048420, −10.08031408269002, −9.952914956288248, −9.148455512534274, −8.920530081813670, −8.536020448175448, −8.058910069914487, −7.553426977511428, −7.155435309975845, −6.535422501513426, −6.117721426738105, −5.684467978587177, −5.182416743036955, −4.286604255523118, −4.010556778793713, −3.258152736163139, −2.854512125767868, −1.963315476395484, −1.423063980609395, −1.141714986096887, −0.5318639173101475, 0.5318639173101475, 1.141714986096887, 1.423063980609395, 1.963315476395484, 2.854512125767868, 3.258152736163139, 4.010556778793713, 4.286604255523118, 5.182416743036955, 5.684467978587177, 6.117721426738105, 6.535422501513426, 7.155435309975845, 7.553426977511428, 8.058910069914487, 8.536020448175448, 8.920530081813670, 9.148455512534274, 9.952914956288248, 10.08031408269002, 10.45431221048420, 11.04075227962598, 11.56513799239339, 11.83474575539879, 12.22080907911534

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