Properties

Degree $2$
Conductor $14450$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 2·3-s + 4-s + 2·6-s − 4·7-s − 8-s + 9-s − 6·11-s − 2·12-s − 2·13-s + 4·14-s + 16-s − 18-s − 4·19-s + 8·21-s + 6·22-s + 2·24-s + 2·26-s + 4·27-s − 4·28-s + 4·31-s − 32-s + 12·33-s + 36-s − 4·37-s + 4·38-s + 4·39-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.15·3-s + 1/2·4-s + 0.816·6-s − 1.51·7-s − 0.353·8-s + 1/3·9-s − 1.80·11-s − 0.577·12-s − 0.554·13-s + 1.06·14-s + 1/4·16-s − 0.235·18-s − 0.917·19-s + 1.74·21-s + 1.27·22-s + 0.408·24-s + 0.392·26-s + 0.769·27-s − 0.755·28-s + 0.718·31-s − 0.176·32-s + 2.08·33-s + 1/6·36-s − 0.657·37-s + 0.648·38-s + 0.640·39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 14450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(14450\)    =    \(2 \cdot 5^{2} \cdot 17^{2}\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{14450} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 14450,\ (\ :1/2),\ -1)\)

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 \( 1 + T \)
5 \( 1 \)
17 \( 1 \)
good3 \( 1 + 2 T + p T^{2} \)
7 \( 1 + 4 T + p T^{2} \)
11 \( 1 + 6 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
19 \( 1 + 4 T + p T^{2} \)
23 \( 1 + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 4 T + p T^{2} \)
41 \( 1 + 6 T + p T^{2} \)
43 \( 1 + 8 T + p T^{2} \)
47 \( 1 + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 + 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 2 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 + p T^{2} \)
89 \( 1 + 6 T + p T^{2} \)
97 \( 1 - 14 T + p T^{2} \)
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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

−16.50514909362706, −15.99170167281328, −15.48664943359084, −15.10881179870668, −14.11823560512770, −13.23447963183461, −12.99611021317329, −12.36175420541600, −11.86528540951495, −11.21454397742475, −10.47985046240228, −10.14511080505373, −9.919103527317289, −8.827854200677093, −8.434721631247959, −7.546258705974526, −6.977654608850467, −6.391263149716337, −5.881655698070916, −5.213150120846833, −4.615165600426554, −3.397567117772909, −2.810908841118525, −2.039886381877294, −0.5623966478004139, 0, 0.5623966478004139, 2.039886381877294, 2.810908841118525, 3.397567117772909, 4.615165600426554, 5.213150120846833, 5.881655698070916, 6.391263149716337, 6.977654608850467, 7.546258705974526, 8.434721631247959, 8.827854200677093, 9.919103527317289, 10.14511080505373, 10.47985046240228, 11.21454397742475, 11.86528540951495, 12.36175420541600, 12.99611021317329, 13.23447963183461, 14.11823560512770, 15.10881179870668, 15.48664943359084, 15.99170167281328, 16.50514909362706

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