↑

Properties

Label 1-496-496.491-r0-0-0
Degree $1$
Conductor $496$
Sign $-0.398 + 0.917i$
Analytic cond. $2.30341$
Root an. cond. $2.30341$
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  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.5i)13-s + 15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s − 23-s + (0.5 + 0.866i)25-s − i·27-s − i·29-s + 33-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (−0.866 − 0.5i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)9-s + (−0.866 − 0.5i)11-s + (−0.866 − 0.5i)13-s + 15-s + (0.5 + 0.866i)17-s + (0.866 − 0.5i)19-s + (0.866 + 0.5i)21-s − 23-s + (0.5 + 0.866i)25-s − i·27-s − i·29-s + 33-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 496 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.398 + 0.917i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(496\)    =    \(2^{4} \cdot 31\)
Sign: $-0.398 + 0.917i$
Analytic conductor: \(2.30341\)
Root analytic conductor: \(2.30341\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{496} (491, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 496,\ (0:\ ),\ -0.398 + 0.917i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1059366132 + 0.1614741959i\)
\(L(\frac12)\) \(\approx\) \(0.1059366132 + 0.1614741959i\)
\(L(1)\) \(\approx\) \(0.5137149890 + 0.01561560377i\)
\(L(1)\) \(\approx\) \(0.5137149890 + 0.01561560377i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
31 \( 1 \)
good3 \( 1 + (-0.866 + 0.5i)T \)
5 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.866 - 0.5i)T \)
13 \( 1 + (-0.866 - 0.5i)T \)
17 \( 1 + (0.5 + 0.866i)T \)
19 \( 1 + (0.866 - 0.5i)T \)
23 \( 1 - T \)
29 \( 1 - iT \)
37 \( 1 + (-0.866 + 0.5i)T \)
41 \( 1 + (0.5 - 0.866i)T \)
43 \( 1 + (0.866 - 0.5i)T \)
47 \( 1 - T \)
53 \( 1 + (0.866 + 0.5i)T \)
59 \( 1 + (-0.866 + 0.5i)T \)
61 \( 1 - iT \)
67 \( 1 + (-0.866 - 0.5i)T \)
71 \( 1 + (-0.5 + 0.866i)T \)
73 \( 1 + (-0.5 + 0.866i)T \)
79 \( 1 + (-0.5 - 0.866i)T \)
83 \( 1 + (0.866 + 0.5i)T \)
89 \( 1 + T \)
97 \( 1 + T \)
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

−23.18454985211101700697010661633, −22.73858019026440646984940625045, −22.0334928300719750006790813757, −21.01599010392316622371296942430, −19.748899859183435955872862835735, −19.008212704721938240020557051362, −18.3578630714161594385437884551, −17.71507344519350081832236342725, −16.26705557645984568924707845046, −16.00282593390837752054737264913, −14.94547287075248658569584229728, −13.889299871223950170810529881889, −12.67571551732689979243253414217, −12.01198331762535753251202586332, −11.541261620206864019962623261254, −10.29262419588019242145067488348, −9.54724390586658032116358303998, −7.9198888955756035203935445098, −7.43711659165454001185647935388, −6.40298913724275010546466068636, −5.42628755135930040290514739045, −4.50670555966278877885038472596, −3.03910057859780085186074328889, −2.062213754536078339393647681402, −0.14286041653270804592858158598, 1.01212751251297122603326234613, 3.1458453724328463380552287048, 3.98602776030341630989142469722, 4.9804646404552909227110079930, 5.78287004583758051914456408960, 7.12213980828946484970937122792, 7.80931473093268737106567061905, 9.06423554028622638344122428408, 10.25847282954116636476768289317, 10.65164346231097407591436437780, 11.85250022958077584556933735133, 12.497376352822221608502610342161, 13.39033091804801767285631527982, 14.686729692112951582786590272126, 15.753122710223483178750955237937, 16.17237525554875349714790733992, 16.99685186889156250721955043044, 17.78594314020753244071819633634, 18.9267905640851902932525693857, 19.85001956262168875278134281690, 20.50919289453185640737893800794, 21.50621532224003227343559290858, 22.421495530245711094009230654029, 23.065163060164494885805581206971, 24.0906589686101485757353437988

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