↑

Properties

Label 1-6223-6223.1054-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.998 - 0.0554i$
Analytic cond. $28.8994$
Root an. cond. $28.8994$
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  + (0.955 + 0.294i)2-s + (−0.733 − 0.680i)3-s + (0.826 + 0.563i)4-s + (0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 − 0.930i)10-s + (−0.900 + 0.433i)11-s + (−0.222 − 0.974i)12-s + (0.955 − 0.294i)13-s + (−0.733 + 0.680i)15-s + (0.365 + 0.930i)16-s + (−0.900 − 0.433i)17-s + (−0.222 + 0.974i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.955 + 0.294i)2-s + (−0.733 − 0.680i)3-s + (0.826 + 0.563i)4-s + (0.0747 − 0.997i)5-s + (−0.5 − 0.866i)6-s + (0.623 + 0.781i)8-s + (0.0747 + 0.997i)9-s + (0.365 − 0.930i)10-s + (−0.900 + 0.433i)11-s + (−0.222 − 0.974i)12-s + (0.955 − 0.294i)13-s + (−0.733 + 0.680i)15-s + (0.365 + 0.930i)16-s + (−0.900 − 0.433i)17-s + (−0.222 + 0.974i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.998 - 0.0554i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1054, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.998 - 0.0554i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.289010831 - 0.06349340841i\)
\(L(\frac12)\) \(\approx\) \(2.289010831 - 0.06349340841i\)
\(L(1)\) \(\approx\) \(1.423775281 - 0.09593887016i\)
\(L(1)\) \(\approx\) \(1.423775281 - 0.09593887016i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.955 + 0.294i)T \)
3 \( 1 + (-0.733 - 0.680i)T \)
5 \( 1 + (0.0747 - 0.997i)T \)
11 \( 1 + (-0.900 + 0.433i)T \)
13 \( 1 + (0.955 - 0.294i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + (-0.5 + 0.866i)T \)
23 \( 1 + (-0.222 + 0.974i)T \)
29 \( 1 + (0.0747 - 0.997i)T \)
31 \( 1 + (-0.733 - 0.680i)T \)
37 \( 1 + (-0.988 - 0.149i)T \)
41 \( 1 + (0.0747 - 0.997i)T \)
43 \( 1 + (0.365 + 0.930i)T \)
47 \( 1 + (0.365 + 0.930i)T \)
53 \( 1 + (-0.222 + 0.974i)T \)
59 \( 1 + (-0.222 + 0.974i)T \)
61 \( 1 + (0.0747 - 0.997i)T \)
67 \( 1 + (0.826 - 0.563i)T \)
71 \( 1 + (0.955 - 0.294i)T \)
73 \( 1 + (0.955 - 0.294i)T \)
79 \( 1 + (0.623 + 0.781i)T \)
83 \( 1 + (0.826 - 0.563i)T \)
89 \( 1 + (0.826 - 0.563i)T \)
97 \( 1 + (0.826 + 0.563i)T \)
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

−17.74237189495137794130610572175, −16.82106423190242142965830622531, −16.01499819299912080742360028106, −15.70678828771478458968495608037, −15.058973764176133132078679722328, −14.44781097596845981275422926001, −13.74109820955278148092257339652, −13.05610021277097858340619273758, −12.45809016192374219287224002759, −11.48921181289853340505300466020, −11.08533801445182523286252316819, −10.53006948338734553240654277200, −10.30218413640297040002330581318, −9.13490432597668580077248900461, −8.4214802661899677337430253906, −7.19533865003247272875655193611, −6.50757584617172310188880795160, −6.30154194227454227777086540390, −5.27436173478061261419985320939, −4.88133917968041843446196723185, −3.797142307225977351232204650700, −3.54391852298512847825221140196, −2.57643027670217997300874400169, −1.896949722559382277385392027871, −0.59652133095810726886967334898, 0.69556173929088895232049888696, 1.88308058210346077272660899941, 2.11622513168051659968492251024, 3.38662863050585172922642407894, 4.28105146149607907643954935713, 4.79128496045300846440552101808, 5.70605392344924345270089160678, 5.8298787283514091698353218627, 6.72145665652934163691379465079, 7.70663486411386620059015318795, 7.88191483483103143179240325459, 8.77994415261720894113087818122, 9.77562715158007134253549925206, 10.8158924058116801756935439853, 11.1259642441218274145588411637, 12.04891297355811648017522383828, 12.518887245926272511111746168, 13.01876038698528181237845442583, 13.62584747530411386953888394113, 13.9835900252577129590694263999, 15.335999212291740100295077346041, 15.672718327941065480953418262847, 16.227859111217647181566560420, 16.97722491658664147774123442083, 17.48172195697050531277675544061

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