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Properties

Label 1-1491-1491.1490-r1-0-0
Degree $1$
Conductor $1491$
Sign $1$
Analytic cond. $160.230$
Root an. cond. $160.230$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 13-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s − 26-s − 29-s + 31-s − 32-s + 34-s + 37-s + 38-s − 40-s − 41-s + 43-s + 44-s − 46-s − 47-s + ⋯
L(s)  = 1  − 2-s + 4-s + 5-s − 8-s − 10-s + 11-s + 13-s + 16-s − 17-s − 19-s + 20-s − 22-s + 23-s + 25-s − 26-s − 29-s + 31-s − 32-s + 34-s + 37-s + 38-s − 40-s − 41-s + 43-s + 44-s − 46-s − 47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1491 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1491 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1491\)    =    \(3 \cdot 7 \cdot 71\)
Sign: $1$
Analytic conductor: \(160.230\)
Root analytic conductor: \(160.230\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{1491} (1490, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 1491,\ (1:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.032721598\)
\(L(\frac12)\) \(\approx\) \(2.032721598\)
\(L(1)\) \(\approx\) \(0.9763202496\)
\(L(1)\) \(\approx\) \(0.9763202496\)

Euler product

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

Imaginary part of the first few zeros on the critical line

−20.46146250304026018732547831344, −19.573578509713560823214029199541, −18.886610340369039557332970594533, −18.13666368778423201145535413653, −17.43532609945437478758693951691, −16.933798274866534610615627353632, −16.22063352545108620536193509719, −15.173503494096832334894604308265, −14.65340705415135426387262329248, −13.50413871400265146215522818422, −12.94335565718873151347547443509, −11.80656543504499494180520767431, −11.03574539889433144521312988512, −10.474510936150074013368641199890, −9.45813373343059461384668233068, −8.96864128439067658380166116921, −8.334596939933070244744331596218, −7.07992637540804087550758305348, −6.39757427677316629592801578000, −5.922267483970132438378317581592, −4.605269718755504051478388992156, −3.44612011975552358339387530808, −2.36676163979979082239351189184, −1.60923750094764345412384583210, −0.74336781637499039065117755217, 0.74336781637499039065117755217, 1.60923750094764345412384583210, 2.36676163979979082239351189184, 3.44612011975552358339387530808, 4.605269718755504051478388992156, 5.922267483970132438378317581592, 6.39757427677316629592801578000, 7.07992637540804087550758305348, 8.334596939933070244744331596218, 8.96864128439067658380166116921, 9.45813373343059461384668233068, 10.474510936150074013368641199890, 11.03574539889433144521312988512, 11.80656543504499494180520767431, 12.94335565718873151347547443509, 13.50413871400265146215522818422, 14.65340705415135426387262329248, 15.173503494096832334894604308265, 16.22063352545108620536193509719, 16.933798274866534610615627353632, 17.43532609945437478758693951691, 18.13666368778423201145535413653, 18.886610340369039557332970594533, 19.573578509713560823214029199541, 20.46146250304026018732547831344

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