↑

Properties

Label 1-4284-4284.1339-r1-0-0
Degree $1$
Conductor $4284$
Sign $0.962 + 0.269i$
Analytic cond. $460.379$
Root an. cond. $460.379$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + i·5-s − i·11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)19-s − i·23-s − 25-s + (0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.866 − 0.5i)37-s + (0.866 + 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s − 55-s + (−0.5 − 0.866i)59-s + ⋯
L(s)  = 1  + i·5-s − i·11-s + (−0.5 + 0.866i)13-s + (−0.5 − 0.866i)19-s − i·23-s − 25-s + (0.866 − 0.5i)29-s + (−0.866 + 0.5i)31-s + (0.866 − 0.5i)37-s + (0.866 + 0.5i)41-s + (−0.5 − 0.866i)43-s + (0.5 − 0.866i)47-s + (0.5 − 0.866i)53-s − 55-s + (−0.5 − 0.866i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4284 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.962 + 0.269i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4284\)    =    \(2^{2} \cdot 3^{2} \cdot 7 \cdot 17\)
Sign: $0.962 + 0.269i$
Analytic conductor: \(460.379\)
Root analytic conductor: \(460.379\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4284} (1339, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4284,\ (1:\ ),\ 0.962 + 0.269i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.735256778 + 0.2384604920i\)
\(L(\frac12)\) \(\approx\) \(1.735256778 + 0.2384604920i\)
\(L(1)\) \(\approx\) \(0.9725103469 + 0.2395201604i\)
\(L(1)\) \(\approx\) \(0.9725103469 + 0.2395201604i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
17 \( 1 \)
good5 \( 1 + iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 + (-0.5 - 0.866i)T \)
23 \( 1 - iT \)
29 \( 1 + (0.866 - 0.5i)T \)
31 \( 1 + (-0.866 + 0.5i)T \)
37 \( 1 + (0.866 - 0.5i)T \)
41 \( 1 + (0.866 + 0.5i)T \)
43 \( 1 + (-0.5 - 0.866i)T \)
47 \( 1 + (0.5 - 0.866i)T \)
53 \( 1 + (0.5 - 0.866i)T \)
59 \( 1 + (-0.5 - 0.866i)T \)
61 \( 1 + (0.866 + 0.5i)T \)
67 \( 1 + (0.5 + 0.866i)T \)
71 \( 1 - iT \)
73 \( 1 + (-0.866 - 0.5i)T \)
79 \( 1 + (-0.866 - 0.5i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.5 - 0.866i)T \)
97 \( 1 + (0.866 - 0.5i)T \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.18432956537811100622678517014, −17.32026673229365442852171955584, −16.7404467542729944663355094998, −16.26560160901674358793679618002, −15.57785281061411121273279193055, −14.69692661583737404022414183062, −14.13287801846752274444726754042, −13.2641029953746125216449680696, −12.650989586278474540699565492752, −12.25348765421279364104848813929, −11.28986632103403771060707840716, −10.62615495594646893489674142438, −9.87247303918282813937670430893, −9.132568983438754162433313513973, −8.300848485084415579338502096055, −8.067151535913076669642151153763, −7.04123263961234525583919930936, −5.94004066362574635132673280263, −5.67129135499375395788715359492, −4.66377543183896219473101683567, −4.09070530580737981372948173029, −3.09652114213011740231951075528, −2.33828417711548082365547989261, −1.1908453993310728783334656180, −0.61458773206310308405820444078, 0.380357942876173476974830365295, 1.79694842241826658185360037069, 2.276500311191056745920473577165, 3.14562038355581777736956347483, 4.05909211808719621277637485762, 4.67705962272275411733092304650, 5.61767013510561630343823380493, 6.48789027018267352479017608749, 7.14871931144557781254701780085, 7.47367441603591847929746719163, 8.587778572195531096548232435651, 9.394373905402824519881889225576, 9.966909271158022617759995966658, 10.63637664702938091489760310060, 11.49208374944759300255062052549, 11.861636281423745789335932702236, 12.84748464387513269115456484565, 13.472377857852951759303140124463, 14.36702699688730343619748302281, 14.71974612234048039455850591875, 15.4590412440957546375527546038, 16.0428652341088544374711096822, 17.057663319766295305697971501518, 17.58402462389736593641553118858, 18.14928495511092226896480944332

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