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Properties

Label 1-77-77.65-r1-0-0
Degree $1$
Conductor $77$
Sign $0.0633 - 0.997i$
Analytic cond. $8.27479$
Root an. cond. $8.27479$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $0.0633 - 0.997i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (65, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ 0.0633 - 0.997i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02398348336 + 0.02250973164i\)
\(L(\frac12)\) \(\approx\) \(0.02398348336 + 0.02250973164i\)
\(L(1)\) \(\approx\) \(0.5961857215 + 0.3965719450i\)
\(L(1)\) \(\approx\) \(0.5961857215 + 0.3965719450i\)

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

−30.94883885679734357989920827522, −30.112557231579447643975463534239, −29.45815231286545183368219289741, −28.280605390604500151153350750481, −27.32733088588348226194496746251, −25.92853661279018772841389024673, −24.23346215169183134371519098836, −23.6068427904832031595421651446, −22.427799554378269105097713501208, −21.862309342021910643441954579802, −20.03578933837683324455379388635, −19.21382844604431397394167601267, −18.381875737866582344845470960874, −17.21721972698921281534505020460, −15.2829557162154334339151841511, −14.196177946329940429022865510179, −13.019018842228865372990556712191, −11.88323204496605895086883380662, −11.13568559224866176033538661932, −9.85719430819394854935732867914, −7.833672125283971141665338417282, −6.52586381500112728642611169265, −5.15032142446298601093268385897, −3.371429583587695836032807087447, −1.935472101595818822224280081007, 0.014001105417621767629151918761, 3.526714227184543689146295785946, 4.75663048434033938965301313099, 5.57482814465451561857889504578, 7.29746186663187685815110455059, 8.67489608376975122755631825544, 9.84598425337698162895725021083, 11.77016631189623617129559075149, 12.5070659667682363250424972010, 14.19867784085992649165321020302, 15.26412804954931839240674065011, 16.39222611395488261382299650023, 16.806492312951544759237361555798, 18.21911805501975771209321067559, 20.17398788587329450401426034724, 21.11876226276270615916127228463, 22.286395443780680080977404764141, 23.127268327916201789439376833727, 24.18727365556074731346939119281, 25.15657102173959804169074014746, 26.66395140509331634760083405414, 27.25100197873369830348981772447, 28.37696214000757825759147930374, 29.6761916142646840996891994435, 31.37327089186588329439347868547

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