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Properties

Label 1-816-816.317-r0-0-0
Degree $1$
Conductor $816$
Sign $-0.990 - 0.139i$
Analytic cond. $3.78948$
Root an. cond. $3.78948$
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  + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)7-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 + 0.707i)19-s + (−0.923 − 0.382i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + (0.923 − 0.382i)31-s − i·35-s + (0.382 + 0.923i)37-s + (0.382 − 0.923i)41-s + (−0.707 − 0.707i)43-s + i·47-s + (−0.707 − 0.707i)49-s + ⋯
L(s)  = 1  + (−0.923 + 0.382i)5-s + (−0.382 + 0.923i)7-s + (−0.382 + 0.923i)11-s + 13-s + (−0.707 + 0.707i)19-s + (−0.923 − 0.382i)23-s + (0.707 − 0.707i)25-s + (−0.923 + 0.382i)29-s + (0.923 − 0.382i)31-s − i·35-s + (0.382 + 0.923i)37-s + (0.382 − 0.923i)41-s + (−0.707 − 0.707i)43-s + i·47-s + (−0.707 − 0.707i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 816 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.139i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(816\)    =    \(2^{4} \cdot 3 \cdot 17\)
Sign: $-0.990 - 0.139i$
Analytic conductor: \(3.78948\)
Root analytic conductor: \(3.78948\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{816} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 816,\ (0:\ ),\ -0.990 - 0.139i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02573711656 + 0.3678447713i\)
\(L(\frac12)\) \(\approx\) \(0.02573711656 + 0.3678447713i\)
\(L(1)\) \(\approx\) \(0.6706907235 + 0.2242228414i\)
\(L(1)\) \(\approx\) \(0.6706907235 + 0.2242228414i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
17 \( 1 \)
good5 \( 1 + (-0.923 + 0.382i)T \)
7 \( 1 + (-0.382 + 0.923i)T \)
11 \( 1 + (-0.382 + 0.923i)T \)
13 \( 1 + T \)
19 \( 1 + (-0.707 + 0.707i)T \)
23 \( 1 + (-0.923 - 0.382i)T \)
29 \( 1 + (-0.923 + 0.382i)T \)
31 \( 1 + (0.923 - 0.382i)T \)
37 \( 1 + (0.382 + 0.923i)T \)
41 \( 1 + (0.382 - 0.923i)T \)
43 \( 1 + (-0.707 - 0.707i)T \)
47 \( 1 + iT \)
53 \( 1 + (-0.707 + 0.707i)T \)
59 \( 1 + (-0.707 - 0.707i)T \)
61 \( 1 + (-0.923 - 0.382i)T \)
67 \( 1 - iT \)
71 \( 1 + (-0.923 + 0.382i)T \)
73 \( 1 + (-0.382 - 0.923i)T \)
79 \( 1 + (0.923 + 0.382i)T \)
83 \( 1 + (-0.707 + 0.707i)T \)
89 \( 1 - iT \)
97 \( 1 + (-0.382 - 0.923i)T \)
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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

−21.69444393366175743213389988725, −20.94834003430753535807507824757, −20.06595060919078955444147735011, −19.51123325787725116942242176659, −18.751970305492383843044088812888, −17.78518941300992356300770163011, −16.73835360876824698665192800177, −16.17172935473693341841367716037, −15.56190456774612911812812142265, −14.501867110078407118752776552317, −13.34485611696174152995083849751, −13.14393190651235873065028868354, −11.82480909607627073446920316396, −11.11998130789203167141736781715, −10.43660864923615173069287864993, −9.25814353069107916035319726016, −8.32985841148266174515301264967, −7.7275943158987923596404807318, −6.66957811466997711852481975711, −5.78414737398628393032563492936, −4.49907125404391508872229278535, −3.81968127149716388507847430996, −2.96915163171877038445686147551, −1.30764341282250273074609312851, −0.17587012484412755312276414573, 1.75629792804265038042228357594, 2.809953553901427760610840840627, 3.783232089691671551155083520359, 4.656266934188919641964091731899, 5.917753060963723388340098059158, 6.59912361400312133088852491297, 7.76497465221792794560440932899, 8.36305965673869867140648261726, 9.380085677850907480882847856134, 10.366949659068887861359681240425, 11.1625813568400880521958518748, 12.187748835720268590065992966917, 12.55416392187418625623230635221, 13.71043753690207026794406082307, 14.798970479256290890730773017683, 15.40650980647278350757878517327, 15.95267745472009245092482049540, 16.91771500157567511015229199007, 18.14422430097422900575651090559, 18.61276879262115682682042902820, 19.2762900763324746093735859195, 20.324281711590894233641567046313, 20.86978141852680675902134434171, 22.04719916440237858412193617223, 22.622651137099700260361553410321

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