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Properties

Label 1-777-777.257-r1-0-0
Degree $1$
Conductor $777$
Sign $0.386 + 0.922i$
Analytic cond. $83.5002$
Root an. cond. $83.5002$
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  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (0.342 − 0.939i)5-s + (0.866 + 0.5i)8-s − 10-s + (−0.5 − 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (0.342 + 0.939i)20-s + (−0.642 + 0.766i)22-s + (0.866 + 0.5i)23-s + (−0.766 − 0.642i)25-s + 26-s + ⋯
L(s)  = 1  + (−0.342 − 0.939i)2-s + (−0.766 + 0.642i)4-s + (0.342 − 0.939i)5-s + (0.866 + 0.5i)8-s − 10-s + (−0.5 − 0.866i)11-s + (−0.342 + 0.939i)13-s + (0.173 − 0.984i)16-s + (0.984 + 0.173i)17-s + (−0.642 − 0.766i)19-s + (0.342 + 0.939i)20-s + (−0.642 + 0.766i)22-s + (0.866 + 0.5i)23-s + (−0.766 − 0.642i)25-s + 26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $0.386 + 0.922i$
Analytic conductor: \(83.5002\)
Root analytic conductor: \(83.5002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (257, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 777,\ (1:\ ),\ 0.386 + 0.922i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1112642980 + 0.07401867928i\)
\(L(\frac12)\) \(\approx\) \(0.1112642980 + 0.07401867928i\)
\(L(1)\) \(\approx\) \(0.6204603524 - 0.3999529379i\)
\(L(1)\) \(\approx\) \(0.6204603524 - 0.3999529379i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
7 \( 1 \)
37 \( 1 \)
good2 \( 1 + (-0.342 - 0.939i)T \)
5 \( 1 + (0.342 - 0.939i)T \)
11 \( 1 + (-0.5 - 0.866i)T \)
13 \( 1 + (-0.342 + 0.939i)T \)
17 \( 1 + (0.984 + 0.173i)T \)
19 \( 1 + (-0.642 - 0.766i)T \)
23 \( 1 + (0.866 + 0.5i)T \)
29 \( 1 + (-0.866 - 0.5i)T \)
31 \( 1 + (-0.866 - 0.5i)T \)
41 \( 1 + (0.939 + 0.342i)T \)
43 \( 1 + iT \)
47 \( 1 + (-0.5 + 0.866i)T \)
53 \( 1 + (0.939 - 0.342i)T \)
59 \( 1 + (-0.342 - 0.939i)T \)
61 \( 1 + (-0.984 + 0.173i)T \)
67 \( 1 + (-0.173 - 0.984i)T \)
71 \( 1 + (-0.173 - 0.984i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (0.342 - 0.939i)T \)
83 \( 1 + (-0.939 + 0.342i)T \)
89 \( 1 + (-0.984 - 0.173i)T \)
97 \( 1 + (-0.866 + 0.5i)T \)
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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

−22.323211076065590360113980372726, −21.30504529538838751881444288217, −20.32696974914637017398063599828, −19.28409363686704871891689376390, −18.488890836029801405442020371490, −18.028914757424922312113299974758, −17.14182981191991294367990298811, −16.44219484703128663476795027681, −15.22947874110109484837299686111, −14.88997905560885919886806370224, −14.17076624896176352042692639877, −13.09778311466387447652103752677, −12.37419873441471179901138202771, −10.78068443339831843351776200073, −10.30976710952389306001321788891, −9.550481968461477012961374526894, −8.45150229392598525811563354703, −7.38907944767042017881545821049, −7.09012643008568822103381449568, −5.7899147747191943210416959987, −5.30162513391081312580839636552, −3.99177245774709108773340801403, −2.80970348978778074339098717332, −1.57302985203758594077124115561, −0.03751167214973469666510588463, 1.03275527812074854583921012358, 1.997846390336220168858997055744, 3.068987103314944877571144135781, 4.17821356687332671751594401906, 5.0197936480286387423464458325, 5.95603281260074776212776376243, 7.46874948001150348152685962660, 8.2878095557794290935792557041, 9.22069309771269092984778289277, 9.61862087562898789285906470224, 10.869378292533202257585843586322, 11.461950464948338260814767525324, 12.48449827497702335498059488212, 13.12008086372336776576129017529, 13.78921211037478993789570630036, 14.819700954908131607082289881787, 16.24038043999025152147579077573, 16.75571873188310316307020774148, 17.43959725779706168683682598330, 18.46902250721836531959920564578, 19.20751202531750810203292206563, 19.79448024208657720898107669851, 20.9539077106063300451339303001, 21.229691626507115400930406600310, 21.88616522113867991325827234662

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