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Properties

Label 1-777-777.383-r1-0-0
Degree $1$
Conductor $777$
Sign $-0.712 - 0.701i$
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  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + 11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (−0.642 − 0.766i)20-s + (−0.984 + 0.173i)22-s − i·23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯
L(s)  = 1  + (−0.984 + 0.173i)2-s + (0.939 − 0.342i)4-s + (−0.342 − 0.939i)5-s + (−0.866 + 0.5i)8-s + (0.5 + 0.866i)10-s + 11-s + (0.642 − 0.766i)13-s + (0.766 − 0.642i)16-s + (−0.984 + 0.173i)17-s + (0.642 − 0.766i)19-s + (−0.642 − 0.766i)20-s + (−0.984 + 0.173i)22-s − i·23-s + (−0.766 + 0.642i)25-s + (−0.5 + 0.866i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 777 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.712 - 0.701i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(777\)    =    \(3 \cdot 7 \cdot 37\)
Sign: $-0.712 - 0.701i$
Analytic conductor: \(83.5002\)
Root analytic conductor: \(83.5002\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{777} (383, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 777,\ (1:\ ),\ -0.712 - 0.701i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3856972733 - 0.9410887815i\)
\(L(\frac12)\) \(\approx\) \(0.3856972733 - 0.9410887815i\)
\(L(1)\) \(\approx\) \(0.6762141855 - 0.2168250913i\)
\(L(1)\) \(\approx\) \(0.6762141855 - 0.2168250913i\)

Euler product

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

−22.18907295157417064268460788458, −21.72079359825236262244483529617, −20.63250648447333947091597832541, −19.769174673559937478221824296226, −19.26727382219325102828795902721, −18.33065367177662604235011442524, −17.89388532361883679151640160642, −16.855176860563426786042717722068, −16.06610593144351818362396802071, −15.33574582728531490046027175836, −14.41346003497705566772071701429, −13.5820170979803533362482466548, −12.12271274776183006987555521788, −11.58245039913925100107708223877, −10.8828342435291228927843704235, −9.996032810651342083784988447899, −9.09870352218465264572691035408, −8.372582663971725431274196908953, −7.14440378523178855733178016076, −6.8051121217423982828682858281, −5.774545795341756100883811380945, −4.01145095065144396171244180955, −3.34666190981072213411164547582, −2.13742040428517466415959214771, −1.17701913080696921199192491612, 0.381252893824589425079790851939, 1.1151837763521029299760076007, 2.30768357079588755464798275804, 3.637885491011749087998244772236, 4.755031495629315186288753560067, 5.8687335381163861855237630145, 6.72987938962603459860366157569, 7.720790452788065060822328660121, 8.68290080280827277022577145968, 9.01092002244827207737235326095, 10.07799037995384414071749386942, 11.088187784309590546189040759389, 11.75971085077289720597213046166, 12.65991503613661100438233468020, 13.62644798505548576303288703574, 14.82326923548380907602156181710, 15.63345222734070293853787293047, 16.21192315736602902044560881368, 17.11919470659763338549768144598, 17.66249741609899431398890635779, 18.57647329213487810150547561536, 19.568267819176943487426853979266, 20.13016082261283318373868744966, 20.605842106836088727723317006999, 21.73648275395548399337812517161

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