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Properties

Label 1-1573-1573.1043-r0-0-0
Degree $1$
Conductor $1573$
Sign $-0.932 - 0.360i$
Analytic cond. $7.30498$
Root an. cond. $7.30498$
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  + (−0.991 + 0.132i)2-s + (0.669 + 0.743i)3-s + (0.964 − 0.263i)4-s + (0.0855 − 0.996i)5-s + (−0.761 − 0.647i)6-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (−0.104 + 0.994i)9-s + (0.0475 + 0.998i)10-s + (0.841 + 0.540i)12-s + (0.516 + 0.856i)14-s + (0.797 − 0.603i)15-s + (0.861 − 0.508i)16-s + (0.272 − 0.962i)17-s + (−0.0285 − 0.999i)18-s + (−0.0665 − 0.997i)19-s + ⋯
L(s)  = 1  + (−0.991 + 0.132i)2-s + (0.669 + 0.743i)3-s + (0.964 − 0.263i)4-s + (0.0855 − 0.996i)5-s + (−0.761 − 0.647i)6-s + (−0.398 − 0.917i)7-s + (−0.921 + 0.389i)8-s + (−0.104 + 0.994i)9-s + (0.0475 + 0.998i)10-s + (0.841 + 0.540i)12-s + (0.516 + 0.856i)14-s + (0.797 − 0.603i)15-s + (0.861 − 0.508i)16-s + (0.272 − 0.962i)17-s + (−0.0285 − 0.999i)18-s + (−0.0665 − 0.997i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1573 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.932 - 0.360i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1573\)    =    \(11^{2} \cdot 13\)
Sign: $-0.932 - 0.360i$
Analytic conductor: \(7.30498\)
Root analytic conductor: \(7.30498\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1573} (1043, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1573,\ (0:\ ),\ -0.932 - 0.360i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06116487836 - 0.3281072292i\)
\(L(\frac12)\) \(\approx\) \(0.06116487836 - 0.3281072292i\)
\(L(1)\) \(\approx\) \(0.6688082026 - 0.05979784704i\)
\(L(1)\) \(\approx\) \(0.6688082026 - 0.05979784704i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad11 \( 1 \)
13 \( 1 \)
good2 \( 1 + (-0.991 + 0.132i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (0.0855 - 0.996i)T \)
7 \( 1 + (-0.398 - 0.917i)T \)
17 \( 1 + (0.272 - 0.962i)T \)
19 \( 1 + (-0.0665 - 0.997i)T \)
23 \( 1 + (-0.995 - 0.0950i)T \)
29 \( 1 + (0.345 - 0.938i)T \)
31 \( 1 + (-0.998 - 0.0570i)T \)
37 \( 1 + (-0.935 + 0.353i)T \)
41 \( 1 + (-0.432 - 0.901i)T \)
43 \( 1 + (-0.327 + 0.945i)T \)
47 \( 1 + (-0.0285 + 0.999i)T \)
53 \( 1 + (-0.870 + 0.491i)T \)
59 \( 1 + (-0.432 + 0.901i)T \)
61 \( 1 + (-0.991 - 0.132i)T \)
67 \( 1 + (0.723 - 0.690i)T \)
71 \( 1 + (-0.532 + 0.846i)T \)
73 \( 1 + (-0.736 + 0.676i)T \)
79 \( 1 + (0.974 - 0.226i)T \)
83 \( 1 + (0.198 + 0.980i)T \)
89 \( 1 + (0.928 + 0.371i)T \)
97 \( 1 + (0.820 - 0.572i)T \)
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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

−20.57717321101487891121959603966, −19.80262763874278620016376096172, −19.1619481986610114658304604768, −18.618129577075414581079315162788, −18.1692273366764561874133461150, −17.46313055327804527890566352291, −16.40982369091138522256053396786, −15.56211042808987306810630659404, −14.825889677495139826923189201885, −14.34699801122152331594974590183, −13.17820721976667847661010234370, −12.25731771293668127028042224307, −11.917986902520339438095840155046, −10.743347062973631896755347971624, −10.069297594044025179553961682512, −9.28461043639768379006301386946, −8.45719886576617562032117816108, −7.87653946257789346098165086286, −6.99452918705658842351541975007, −6.30807094019054179558833909795, −5.70553644401863440655969859043, −3.50940577481039989300676721251, −3.278694409088571911618901937207, −1.994674706052956026585004816819, −1.77845914873523344681715989280, 0.14508060170222571219746599167, 1.33741941050044897691704292734, 2.40749798477620562513019922355, 3.37213988844802947652271581120, 4.37337921990255705954105059164, 5.17489197997232872039357639459, 6.250462645339806883127746104812, 7.36752556534754136120556105433, 7.91721700009897528494366169095, 8.81321270257951769205105452125, 9.43384930338307592392907018482, 9.95345686080668918204443141019, 10.75610739009316274564933433173, 11.60075316455517718664523261672, 12.59070401019011329705922519016, 13.63243123649902851246812208727, 14.11243959513205480165737897928, 15.2936544163070807613292423791, 15.92633489374756868672876944691, 16.39961566371632785361578902776, 17.09263001958717884505503866751, 17.75261829837646308249205746450, 18.88702884357747652840438415318, 19.627822646021282918701018824043, 20.17191438721074202482538565443

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