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Properties

Label 1-1480-1480.867-r0-0-0
Degree $1$
Conductor $1480$
Sign $-0.965 - 0.259i$
Analytic cond. $6.87309$
Root an. cond. $6.87309$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (−0.766 + 0.642i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.984 − 0.173i)33-s + (0.766 + 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯
L(s)  = 1  + (−0.642 − 0.766i)3-s + (−0.342 + 0.939i)7-s + (−0.173 + 0.984i)9-s + (−0.5 + 0.866i)11-s + (−0.984 + 0.173i)13-s + (0.984 + 0.173i)17-s + (−0.766 + 0.642i)19-s + (0.939 − 0.342i)21-s + (0.866 − 0.5i)23-s + (0.866 − 0.5i)27-s + (−0.5 + 0.866i)29-s − 31-s + (0.984 − 0.173i)33-s + (0.766 + 0.642i)39-s + (0.173 + 0.984i)41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1480 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.259i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1480\)    =    \(2^{3} \cdot 5 \cdot 37\)
Sign: $-0.965 - 0.259i$
Analytic conductor: \(6.87309\)
Root analytic conductor: \(6.87309\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1480} (867, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1480,\ (0:\ ),\ -0.965 - 0.259i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.001803346862 + 0.01367371111i\)
\(L(\frac12)\) \(\approx\) \(0.001803346862 + 0.01367371111i\)
\(L(1)\) \(\approx\) \(0.6360809816 + 0.005869125246i\)
\(L(1)\) \(\approx\) \(0.6360809816 + 0.005869125246i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
37 \( 1 \)
good3 \( 1 + (-0.642 - 0.766i)T \)
7 \( 1 + (-0.342 + 0.939i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.984 + 0.173i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.766 + 0.642i)T \)
23 \( 1 + (0.866 - 0.5i)T \)
29 \( 1 + (-0.5 + 0.866i)T \)
31 \( 1 - T \)
41 \( 1 + (0.173 + 0.984i)T \)
43 \( 1 - iT \)
47 \( 1 + (-0.866 + 0.5i)T \)
53 \( 1 + (-0.342 - 0.939i)T \)
59 \( 1 + (0.939 - 0.342i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.342 - 0.939i)T \)
71 \( 1 + (-0.766 + 0.642i)T \)
73 \( 1 - iT \)
79 \( 1 + (-0.939 - 0.342i)T \)
83 \( 1 + (-0.984 - 0.173i)T \)
89 \( 1 + (0.939 - 0.342i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
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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

−20.38162228915549847982918294294, −19.49498054275428226974337150649, −18.86426631205967905372372776166, −17.74781646875963579175362924046, −17.0667914237049449639305655937, −16.60526717164078015749192338624, −15.89136523109394184977522051966, −15.00046024442155874694569421190, −14.3589735172663453675041645741, −13.30369696888448091397093575495, −12.696898996953361939704402931093, −11.620067121338353933493435532740, −10.99239860058851598773198203469, −10.270572156443632278151203192429, −9.66615024571286355119549474145, −8.792768743850055571400220191205, −7.6293132680899901456248481829, −6.963783504210203346528052965627, −5.89236438045596834065384875404, −5.24447801256319770276772698828, −4.348716891300456432244444044263, −3.51364221656086796865596528198, −2.7038277468114946176563016951, −1.02545171110361110188022981791, −0.0063704420964365463123180568, 1.636624486434251565045587561326, 2.28067769122951466579069607259, 3.29071949461946493183000930268, 4.77929747731488450062258956810, 5.30514770019903503866695576427, 6.17487961500400491135368922197, 7.00929369639632090356555053576, 7.69790714489716460830924391918, 8.57636680551127522440834284980, 9.60934563560129520640585560962, 10.33255054903430616042635659839, 11.2586380719088890362610553655, 12.15112865518868979914256266029, 12.67037256399706843298380434614, 13.026693700403614611588749558863, 14.55746285181581989640103220110, 14.7596468561677917000625706934, 15.9666210339699618821483068590, 16.67891394296011095169748258542, 17.31842477096708300928404011004, 18.20517608091882328355919780619, 18.76124738292038980642490743043, 19.29982044152223834533438499859, 20.22560405026550161174626605302, 21.20346471578492084966971855153

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