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Properties

Label 1-77-77.54-r0-0-0
Degree $1$
Conductor $77$
Sign $0.605 + 0.795i$
Analytic cond. $0.357586$
Root an. cond. $0.357586$
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 + (0.5 + 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + 13-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s − 20-s + ⋯
L(s)  = 1  + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (0.5 + 0.866i)5-s + 6-s − 8-s + (−0.5 − 0.866i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)12-s + 13-s + 15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s + (0.5 − 0.866i)18-s + (−0.5 − 0.866i)19-s − 20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.605 + 0.795i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (54, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ 0.605 + 0.795i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.221059764 + 0.6053126052i\)
\(L(\frac12)\) \(\approx\) \(1.221059764 + 0.6053126052i\)
\(L(1)\) \(\approx\) \(1.331448748 + 0.4762539847i\)
\(L(1)\) \(\approx\) \(1.331448748 + 0.4762539847i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (0.5 + 0.866i)T \)
3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
13 \( 1 + T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 - T \)
31 \( 1 + (0.5 - 0.866i)T \)
37 \( 1 + (-0.5 - 0.866i)T \)
41 \( 1 + T \)
43 \( 1 - T \)
47 \( 1 + (0.5 + 0.866i)T \)
53 \( 1 + (-0.5 + 0.866i)T \)
59 \( 1 + (0.5 - 0.866i)T \)
61 \( 1 + (-0.5 - 0.866i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.5 + 0.866i)T \)
83 \( 1 + T \)
89 \( 1 + (0.5 + 0.866i)T \)
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

−31.429733020803012651912806646, −30.08121609275654623305344295294, −28.89102990816694583204549170953, −28.00394136861537638706290022805, −27.200378372361830236230823314116, −25.73088061466905910785663156650, −24.63031931661766109454417989562, −23.26163460350050812042066791751, −22.10051973045745786445429006155, −21.040915409555038347622366172005, −20.55399779341997317414590520228, −19.48160808208894170212581985758, −18.031601801976409932177716772739, −16.478131268436050421499555702881, −15.35443441671282396118523702390, −13.97003024876400545194576256467, −13.258557780549218475262547542273, −11.768037074581359112925633329172, −10.46872315846050996893627407209, −9.42474220542436190438340543265, −8.49181088360221259313042737562, −5.82486850086385210296555087717, −4.69326035192166651314045530225, −3.49914479518713627012347491644, −1.835647058004524258086670431736, 2.37795666442330338369549924462, 3.831861271522996116507754396911, 5.98696871515592190548818088847, 6.69971362846210885761857272603, 7.97459201833555979780154168332, 9.13939337893263186651750096026, 11.10083931861808648159247460919, 12.742146745782070346916488042407, 13.58186053630132600154934974532, 14.53496217676486107777970987644, 15.489733326569744949494233017454, 17.20366651319625459894101489391, 18.07963348073779674641490339958, 19.02035033674705711934176573120, 20.661046270118752709478913054013, 21.88742092489719424792737960858, 22.97510468538297688463406439944, 23.96756659173325161819309502438, 24.94389531559146119719437032702, 26.08172571574641486202970140465, 26.31123182651436464654631150745, 28.26498418973395830045133712201, 29.91628108938708949581390455286, 30.36902005191543815619956373450, 31.35228280808275001732154612555

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