Properties

Label 2-58443-1.1-c1-0-8
Degree $2$
Conductor $58443$
Sign $-1$
Analytic cond. $466.669$
Root an. cond. $21.6025$
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  + 3-s − 2·4-s + 3·5-s − 7-s + 9-s − 2·12-s − 2·13-s + 3·15-s + 4·16-s − 3·17-s − 2·19-s − 6·20-s − 21-s − 23-s + 4·25-s + 27-s + 2·28-s + 8·31-s − 3·35-s − 2·36-s + 2·37-s − 2·39-s + 6·41-s − 5·43-s + 3·45-s + 9·47-s + 4·48-s + ⋯
L(s)  = 1  + 0.577·3-s − 4-s + 1.34·5-s − 0.377·7-s + 1/3·9-s − 0.577·12-s − 0.554·13-s + 0.774·15-s + 16-s − 0.727·17-s − 0.458·19-s − 1.34·20-s − 0.218·21-s − 0.208·23-s + 4/5·25-s + 0.192·27-s + 0.377·28-s + 1.43·31-s − 0.507·35-s − 1/3·36-s + 0.328·37-s − 0.320·39-s + 0.937·41-s − 0.762·43-s + 0.447·45-s + 1.31·47-s + 0.577·48-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 58443 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58443 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(58443\)    =    \(3 \cdot 7 \cdot 11^{2} \cdot 23\)
Sign: $-1$
Analytic conductor: \(466.669\)
Root analytic conductor: \(21.6025\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{58443} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 58443,\ (\ :1/2),\ -1)\)

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)$
bad3 \( 1 - T \)
7 \( 1 + T \)
11 \( 1 \)
23 \( 1 + T \)
good2 \( 1 + p T^{2} \)
5 \( 1 - 3 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 3 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
29 \( 1 + p T^{2} \)
31 \( 1 - 8 T + p T^{2} \)
37 \( 1 - 2 T + p T^{2} \)
41 \( 1 - 6 T + p T^{2} \)
43 \( 1 + 5 T + p T^{2} \)
47 \( 1 - 9 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 - 9 T + p T^{2} \)
61 \( 1 + 8 T + p T^{2} \)
67 \( 1 + 13 T + p T^{2} \)
71 \( 1 + 12 T + p T^{2} \)
73 \( 1 + 14 T + p T^{2} \)
79 \( 1 + 8 T + p T^{2} \)
83 \( 1 - 3 T + p T^{2} \)
89 \( 1 - 9 T + p T^{2} \)
97 \( 1 + 4 T + p T^{2} \)
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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

−14.58379987114754, −13.95752544247532, −13.54290280573444, −13.19736464176036, −12.93345114274345, −12.11593978442951, −11.78135688064209, −10.69421399639032, −10.26833358486326, −9.990579611325523, −9.387742878761569, −8.948169631332744, −8.667838932000603, −7.915137369032592, −7.330952879501564, −6.661593810146156, −5.980185402322794, −5.721017640136938, −4.845418106926276, −4.418008509697495, −3.880550200977036, −2.875150634484994, −2.577287279367388, −1.771793171197445, −1.011287383874318, 0, 1.011287383874318, 1.771793171197445, 2.577287279367388, 2.875150634484994, 3.880550200977036, 4.418008509697495, 4.845418106926276, 5.721017640136938, 5.980185402322794, 6.661593810146156, 7.330952879501564, 7.915137369032592, 8.667838932000603, 8.948169631332744, 9.387742878761569, 9.990579611325523, 10.26833358486326, 10.69421399639032, 11.78135688064209, 12.11593978442951, 12.93345114274345, 13.19736464176036, 13.54290280573444, 13.95752544247532, 14.58379987114754

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