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Properties

Label 1-1027-1027.789-r1-0-0
Degree $1$
Conductor $1027$
Sign $0.252 - 0.967i$
Analytic cond. $110.366$
Root an. cond. $110.366$
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  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s − 12-s − 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)4-s + 5-s + (0.5 − 0.866i)6-s + (0.5 − 0.866i)7-s + 8-s + (−0.5 + 0.866i)9-s + (−0.5 − 0.866i)10-s + (−0.5 − 0.866i)11-s − 12-s − 14-s + (0.5 + 0.866i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + 18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1027 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.252 - 0.967i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1027\)    =    \(13 \cdot 79\)
Sign: $0.252 - 0.967i$
Analytic conductor: \(110.366\)
Root analytic conductor: \(110.366\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1027} (789, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1027,\ (1:\ ),\ 0.252 - 0.967i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.789790339 - 1.382493301i\)
\(L(\frac12)\) \(\approx\) \(1.789790339 - 1.382493301i\)
\(L(1)\) \(\approx\) \(1.134315737 - 0.3117477386i\)
\(L(1)\) \(\approx\) \(1.134315737 - 0.3117477386i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad13 \( 1 \)
79 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.5 + 0.866i)T \)
5 \( 1 + T \)
7 \( 1 + (0.5 - 0.866i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.5 + 0.866i)T \)
31 \( 1 + T \)
37 \( 1 + (0.5 + 0.866i)T \)
41 \( 1 + (0.5 + 0.866i)T \)
43 \( 1 + (0.5 - 0.866i)T \)
47 \( 1 - T \)
53 \( 1 - T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (0.5 - 0.866i)T \)
67 \( 1 + (-0.5 - 0.866i)T \)
71 \( 1 + (0.5 - 0.866i)T \)
73 \( 1 + T \)
83 \( 1 + T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.5 + 0.866i)T \)
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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

−21.41482489697261847298719473253, −20.83566104107794429918205692896, −19.61051663148635759134349831190, −19.113385932905586561734425915123, −18.16655873653685297058444306982, −17.57913496436403965944536988820, −17.414871171985021879110011962701, −15.97469577780219722272438703936, −15.07778219914946303361814225361, −14.629656892248478915808645296629, −13.75194967092579549725073065312, −13.04958503384436498660043417481, −12.275749186514008946883952163625, −11.04876137439383280047592092210, −9.90846758324527026901669662393, −9.346881135707470706787284004960, −8.43135245443983453405244004780, −7.85884563673110957571223024522, −6.89146017160868379325805373134, −6.05098610501039089102294952551, −5.48495984711072600214823673857, −4.37725810215353479586591150008, −2.57517703348446871365883306578, −1.963705026551594020484771748355, −1.02361509296937203930626443028, 0.56459407830679282688788838814, 1.692758077205795011579785521163, 2.71358188448537276929938924066, 3.419845477607792301320002585900, 4.53173675269345580203724937254, 5.169443784800650507719848631390, 6.50659613684769846082029735741, 7.97340577136041616550040679223, 8.3044827856216847579642605128, 9.41052357623254213955793231327, 10.05280907210437735263789865744, 10.621179549908285531780839139759, 11.25076838648658357127020740902, 12.46443099958007499409377397190, 13.50695098553085687281521246279, 13.9830361368520331958611924719, 14.58168060868351783589267731549, 16.14226594734704346770352758979, 16.58527361398285407960172440875, 17.31304587029646719813196800009, 18.23137757409994706058595649398, 18.9078843061918440455586437278, 19.903987002450951856665068516081, 20.74707331645214758144802931555, 20.91487464193168659216578909236

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