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Properties

Label 1-77-77.30-r1-0-0
Degree $1$
Conductor $77$
Sign $-0.180 - 0.983i$
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  + (0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (−0.913 + 0.406i)18-s + (−0.913 − 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯
L(s)  = 1  + (0.978 − 0.207i)2-s + (−0.104 − 0.994i)3-s + (0.913 − 0.406i)4-s + (0.669 − 0.743i)5-s + (−0.309 − 0.951i)6-s + (0.809 − 0.587i)8-s + (−0.978 + 0.207i)9-s + (0.5 − 0.866i)10-s + (−0.5 − 0.866i)12-s + (−0.309 + 0.951i)13-s + (−0.809 − 0.587i)15-s + (0.669 − 0.743i)16-s + (0.978 + 0.207i)17-s + (−0.913 + 0.406i)18-s + (−0.913 − 0.406i)19-s + (0.309 − 0.951i)20-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.180 - 0.983i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(77\)    =    \(7 \cdot 11\)
Sign: $-0.180 - 0.983i$
Analytic conductor: \(8.27479\)
Root analytic conductor: \(8.27479\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{77} (30, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 77,\ (1:\ ),\ -0.180 - 0.983i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.982918219 - 2.379334336i\)
\(L(\frac12)\) \(\approx\) \(1.982918219 - 2.379334336i\)
\(L(1)\) \(\approx\) \(1.696385461 - 1.085870459i\)
\(L(1)\) \(\approx\) \(1.696385461 - 1.085870459i\)

Euler product

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

−31.66751722092153046954857677333, −30.16158967301073086765824309242, −29.51785668545654427934563653919, −28.14069691409525923645728832461, −26.833257315904888625904705012128, −25.72687945384453555267381649576, −25.03537565035009650253969964705, −23.26985669014996403883312299059, −22.58977457213228975525401910056, −21.550266206311522164246874808112, −20.92705709004308937705103916806, −19.56476019048333802267959089894, −17.68430056652823366648016589234, −16.66793940159920873800222687653, −15.372174663556649106104635819268, −14.63266616501654574015981683679, −13.58034143929872775995276260861, −12.04311386732827519126813852915, −10.74141316426056731724146373116, −9.87203944276821932589631634301, −7.888566441547385247794402871486, −6.20324202906335945289341455566, −5.30447187517561070304911984402, −3.76483299075739646201872476114, −2.571801943120830993350832107060, 1.275101530942613042093447257060, 2.53177169170176872069930537087, 4.591119461609039246781800379805, 5.88322495292309912502040971885, 6.90278313893999812356310499912, 8.55336577132694936918794215858, 10.31157489868910884833406952140, 11.91213041517135064800365682424, 12.60173641468955730433382555328, 13.69748866685658727089241720852, 14.50106186081454741788685633719, 16.321278573799957178976912195443, 17.26522616805924533129502392873, 18.816947794674268378341872549152, 19.841040384554893281840409553987, 20.99520476730592580262047285656, 21.96694840846171679351245196897, 23.37897919364672324150503283050, 24.053782746180898172652466184240, 25.005230466069619255015344324396, 25.85220895833616546940043963926, 28.08917903649864519079976769213, 28.8860515475806935832205476048, 29.71394914743614022719603663535, 30.640000669415047221970406184334

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