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Properties

Label 1-783-783.515-r1-0-0
Degree $1$
Conductor $783$
Sign $0.962 - 0.271i$
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  + (0.995 − 0.0995i)2-s + (0.980 − 0.198i)4-s + (−0.270 − 0.962i)5-s + (0.980 + 0.198i)7-s + (0.955 − 0.294i)8-s + (−0.365 − 0.930i)10-s + (0.921 − 0.388i)11-s + (0.542 + 0.840i)13-s + (0.995 + 0.0995i)14-s + (0.921 − 0.388i)16-s + (−0.5 + 0.866i)17-s + (0.988 + 0.149i)19-s + (−0.456 − 0.889i)20-s + (0.878 − 0.478i)22-s + (0.411 + 0.911i)23-s + ⋯
L(s)  = 1  + (0.995 − 0.0995i)2-s + (0.980 − 0.198i)4-s + (−0.270 − 0.962i)5-s + (0.980 + 0.198i)7-s + (0.955 − 0.294i)8-s + (−0.365 − 0.930i)10-s + (0.921 − 0.388i)11-s + (0.542 + 0.840i)13-s + (0.995 + 0.0995i)14-s + (0.921 − 0.388i)16-s + (−0.5 + 0.866i)17-s + (0.988 + 0.149i)19-s + (−0.456 − 0.889i)20-s + (0.878 − 0.478i)22-s + (0.411 + 0.911i)23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 783 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 - 0.271i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(783\)    =    \(3^{3} \cdot 29\)
Sign: $0.962 - 0.271i$
Analytic conductor: \(84.1450\)
Root analytic conductor: \(84.1450\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{783} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 783,\ (1:\ ),\ 0.962 - 0.271i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(5.576807365 - 0.7702859806i\)
\(L(\frac12)\) \(\approx\) \(5.576807365 - 0.7702859806i\)
\(L(1)\) \(\approx\) \(2.447328534 - 0.3203113259i\)
\(L(1)\) \(\approx\) \(2.447328534 - 0.3203113259i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
29 \( 1 \)
good2 \( 1 + (0.995 - 0.0995i)T \)
5 \( 1 + (-0.270 - 0.962i)T \)
7 \( 1 + (0.980 + 0.198i)T \)
11 \( 1 + (0.921 - 0.388i)T \)
13 \( 1 + (0.542 + 0.840i)T \)
17 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (0.988 + 0.149i)T \)
23 \( 1 + (0.411 + 0.911i)T \)
31 \( 1 + (0.969 + 0.246i)T \)
37 \( 1 + (0.733 + 0.680i)T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.270 + 0.962i)T \)
47 \( 1 + (-0.797 - 0.603i)T \)
53 \( 1 + (0.900 - 0.433i)T \)
59 \( 1 + (-0.173 + 0.984i)T \)
61 \( 1 + (-0.980 - 0.198i)T \)
67 \( 1 + (-0.124 - 0.992i)T \)
71 \( 1 + (-0.955 - 0.294i)T \)
73 \( 1 + (-0.0747 - 0.997i)T \)
79 \( 1 + (-0.542 + 0.840i)T \)
83 \( 1 + (0.853 - 0.521i)T \)
89 \( 1 + (0.826 + 0.563i)T \)
97 \( 1 + (-0.878 - 0.478i)T \)
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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

−22.357016859707276041208240086669, −21.568218353912551674471423858070, −20.34649107721170870464287841135, −20.294926081814679381653999955868, −19.01283449093111253960206761586, −18.028852779777035161543205546405, −17.3799234113108062390921691451, −16.248403013504688305058177508798, −15.362537485723406064088697199307, −14.79816674017194895767242012225, −14.05221590682378212906453331236, −13.442093788023368497963933730686, −12.16598623461760623312503524134, −11.51824571045089619789370887595, −10.913635791896799750257797571699, −10.01679348076830755765554995988, −8.55949360463689368135474399271, −7.54200239224837332289861231751, −6.9565059448007844063087545584, −6.02902874655558445108410428284, −4.956598412117161232490356021485, −4.14365713205652902873129254438, −3.18099119911046423967339307274, −2.290876428528440950268737933366, −1.01618814084187457845217588507, 1.26319240575718457878697647329, 1.6449059390061499220630533897, 3.26043066215793230624566823440, 4.20172434743234763990662274885, 4.8240610381464059886374133860, 5.77396344842324455074375085286, 6.65431353146159489610209208554, 7.83861584221542997482898282809, 8.60524478425534954870243716218, 9.58136761260694487235613178776, 10.93312823625380753295954285358, 11.78602195869236889465611564447, 11.926922954790487147990118985840, 13.33269529237230639998705317874, 13.71273710398851175571893987698, 14.77589112156038235488961117317, 15.40443620313775414493948331716, 16.42598772692596951034871963938, 16.924517681764546183348623615352, 17.998651646736001917224174927200, 19.25874101043860521904462684469, 19.84246569215315550563471958335, 20.66934402537522775830991471292, 21.40310036715916907168916957111, 21.841314976589161994985266107697

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