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Properties

Label 1-783-783.158-r1-0-0
Degree $1$
Conductor $783$
Sign $0.542 + 0.840i$
Analytic cond. $84.1450$
Root an. cond. $84.1450$
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  + (−0.0249 + 0.999i)2-s + (−0.998 − 0.0498i)4-s + (0.318 + 0.947i)5-s + (−0.998 + 0.0498i)7-s + (0.0747 − 0.997i)8-s + (−0.955 + 0.294i)10-s + (0.995 + 0.0995i)11-s + (−0.969 + 0.246i)13-s + (−0.0249 − 0.999i)14-s + (0.995 + 0.0995i)16-s + (−0.5 − 0.866i)17-s + (0.733 + 0.680i)19-s + (−0.270 − 0.962i)20-s + (−0.124 + 0.992i)22-s + (−0.878 − 0.478i)23-s + ⋯
L(s)  = 1  + (−0.0249 + 0.999i)2-s + (−0.998 − 0.0498i)4-s + (0.318 + 0.947i)5-s + (−0.998 + 0.0498i)7-s + (0.0747 − 0.997i)8-s + (−0.955 + 0.294i)10-s + (0.995 + 0.0995i)11-s + (−0.969 + 0.246i)13-s + (−0.0249 − 0.999i)14-s + (0.995 + 0.0995i)16-s + (−0.5 − 0.866i)17-s + (0.733 + 0.680i)19-s + (−0.270 − 0.962i)20-s + (−0.124 + 0.992i)22-s + (−0.878 − 0.478i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.542 + 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.542 + 0.840i$
Analytic conductor: \(84.1450\)
Root analytic conductor: \(84.1450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (158, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 783,\ (1:\ ),\ 0.542 + 0.840i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.114047107 + 0.6067633311i\)
\(L(\frac12)\) \(\approx\) \(1.114047107 + 0.6067633311i\)
\(L(1)\) \(\approx\) \(0.7308623915 + 0.4693520753i\)
\(L(1)\) \(\approx\) \(0.7308623915 + 0.4693520753i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (-0.0249 + 0.999i)T \)
5 \( 1 + (0.318 + 0.947i)T \)
7 \( 1 + (-0.998 + 0.0498i)T \)
11 \( 1 + (0.995 + 0.0995i)T \)
13 \( 1 + (-0.969 + 0.246i)T \)
17 \( 1 + (-0.5 - 0.866i)T \)
19 \( 1 + (0.733 + 0.680i)T \)
23 \( 1 + (-0.878 - 0.478i)T \)
31 \( 1 + (0.661 - 0.749i)T \)
37 \( 1 + (-0.826 - 0.563i)T \)
41 \( 1 + (0.766 - 0.642i)T \)
43 \( 1 + (0.318 - 0.947i)T \)
47 \( 1 + (-0.583 + 0.811i)T \)
53 \( 1 + (-0.623 - 0.781i)T \)
59 \( 1 + (0.939 + 0.342i)T \)
61 \( 1 + (0.998 - 0.0498i)T \)
67 \( 1 + (-0.411 + 0.911i)T \)
71 \( 1 + (-0.0747 - 0.997i)T \)
73 \( 1 + (-0.365 - 0.930i)T \)
79 \( 1 + (0.969 + 0.246i)T \)
83 \( 1 + (0.797 - 0.603i)T \)
89 \( 1 + (-0.988 + 0.149i)T \)
97 \( 1 + (0.124 + 0.992i)T \)
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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

−22.018981219384801784241213458882, −21.24213270911398950290186830264, −20.11468213084304790689285272696, −19.71556311701416069174562878603, −19.2473371691081525208712863350, −17.84743981143524739732507072469, −17.347704764631384150858801895298, −16.55165650722974895567640163130, −15.58222373486029682535436098328, −14.36011134703675462465214386022, −13.57213500895075441295716998824, −12.82688682762164892294613580917, −12.20818768674288304435994318083, −11.46416972554433443420222061007, −10.16935911106861674708110543965, −9.632059937850210104937193394974, −8.96162694759521447426104167787, −8.05218279830795342432598565207, −6.67752897345760775239869769192, −5.64261116873990327612306675234, −4.67168242211352416356196620477, −3.82198759170624143508470119062, −2.80769190934075738414112476289, −1.67444670630456208257754916681, −0.69409289347985702212475910778, 0.43970662219939316309201251856, 2.20544806802359388568417304411, 3.38278053449161125215892528293, 4.246033422632753507506212591729, 5.51593739011159604000819092089, 6.35988005775455532286540660694, 6.95102417415318397938093982167, 7.66988716133855088561216572226, 9.04870441837668350427580805078, 9.66802243673045554773668216249, 10.27050270046555607320185346080, 11.70735734196297918019725779066, 12.50207378402077455001449444368, 13.670108048798003297226676981741, 14.18338396381668159184944955206, 14.88590819346670922256442732579, 15.8614360228016565924088553618, 16.46565573415683851173240321809, 17.444895261141772933289864836266, 18.01295052061592405755082720977, 19.09253319520568450539941851746, 19.37678754785878628373878992150, 20.69830571033375786318335325593, 22.0606498114529829997805763455, 22.45040966049896144930963540922

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