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Properties

Label 1-77-77.9-r0-0-0
Degree $1$
Conductor $77$
Sign $-0.410 + 0.911i$
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  + (−0.104 + 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.809 − 0.587i)20-s + ⋯
L(s)  = 1  + (−0.104 + 0.994i)2-s + (0.669 + 0.743i)3-s + (−0.978 − 0.207i)4-s + (0.913 + 0.406i)5-s + (−0.809 + 0.587i)6-s + (0.309 − 0.951i)8-s + (−0.104 + 0.994i)9-s + (−0.5 + 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.809 − 0.587i)13-s + (0.309 + 0.951i)15-s + (0.913 + 0.406i)16-s + (−0.104 − 0.994i)17-s + (−0.978 − 0.207i)18-s + (−0.978 + 0.207i)19-s + (−0.809 − 0.587i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.410 + 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.410 + 0.911i$
Analytic conductor: \(0.357586\)
Root analytic conductor: \(0.357586\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (0:\ ),\ -0.410 + 0.911i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.5956334762 + 0.9216429538i\)
\(L(\frac12)\) \(\approx\) \(0.5956334762 + 0.9216429538i\)
\(L(1)\) \(\approx\) \(0.8676961069 + 0.7641745552i\)
\(L(1)\) \(\approx\) \(0.8676961069 + 0.7641745552i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
11 \( 1 \)
good2 \( 1 + (-0.104 + 0.994i)T \)
3 \( 1 + (0.669 + 0.743i)T \)
5 \( 1 + (0.913 + 0.406i)T \)
13 \( 1 + (-0.809 - 0.587i)T \)
17 \( 1 + (-0.104 - 0.994i)T \)
19 \( 1 + (-0.978 + 0.207i)T \)
23 \( 1 + (-0.5 - 0.866i)T \)
29 \( 1 + (0.309 + 0.951i)T \)
31 \( 1 + (0.913 - 0.406i)T \)
37 \( 1 + (0.669 - 0.743i)T \)
41 \( 1 + (0.309 - 0.951i)T \)
43 \( 1 + T \)
47 \( 1 + (-0.978 + 0.207i)T \)
53 \( 1 + (0.913 - 0.406i)T \)
59 \( 1 + (-0.978 - 0.207i)T \)
61 \( 1 + (0.913 + 0.406i)T \)
67 \( 1 + (-0.5 + 0.866i)T \)
71 \( 1 + (-0.809 + 0.587i)T \)
73 \( 1 + (-0.978 - 0.207i)T \)
79 \( 1 + (-0.104 + 0.994i)T \)
83 \( 1 + (-0.809 + 0.587i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (-0.809 - 0.587i)T \)
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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

−30.70664648384572731116160276862, −29.7894778614515192892214026089, −29.05877510276049425518268090140, −28.047763402004510485256093265725, −26.53143567744537204719028207510, −25.70919276358779005627704782380, −24.46075428979846108219350892390, −23.389745838365380086255614835091, −21.74429271401698563364630471746, −21.09108326493364476911662267100, −19.822789513509559260369358682575, −19.121954441351829675048392651079, −17.811775033332159279237994723610, −17.0806289538135755987823294932, −14.773578955526280501713256707846, −13.693402431179139536437921327578, −12.89192617804966489701676045352, −11.85046229577707308855801031195, −10.12509242221868070317779543925, −9.144053508888540312574143194757, −8.04645232483687150235638968971, −6.2036100536532354434564028196, −4.38359373003624553415255489328, −2.6072534207345332465775049164, −1.560734045844362448759627019972, 2.63246400368537343570482040109, 4.45567490399429252483318831660, 5.686059566119153231509162336737, 7.1698320186905603412186746685, 8.54836105599322430033443532328, 9.664570789924731707619360331147, 10.49924691379127745090794619333, 12.92604332417501172100032129097, 14.148151928713256942027800453695, 14.7473388718454776197515841696, 15.9726447285893869301126831207, 17.06511523540811457951726690824, 18.16043402447972433584158880230, 19.42249536671613540438134301046, 20.88210281599245899490332384987, 22.01920516348369056300930124870, 22.74958518257749446903451692194, 24.51290079164217201935190843174, 25.26393789369063848316239544004, 26.134314163992792849642907900017, 27.005888576444419094454476543482, 27.974643805291777989564889177501, 29.50633087787813592264414110859, 30.86687288234043900866547074907, 32.03973890832587879666578490375

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