↑

Properties

Label 1-6223-6223.4871-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.933 - 0.358i$
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  + 2-s + (−0.661 − 0.749i)3-s + 4-s + (−0.733 + 0.680i)5-s + (−0.661 − 0.749i)6-s + 8-s + (−0.124 + 0.992i)9-s + (−0.733 + 0.680i)10-s + (0.456 + 0.889i)11-s + (−0.661 − 0.749i)12-s + (0.969 + 0.246i)13-s + (0.995 + 0.0995i)15-s + 16-s + (0.939 + 0.342i)17-s + (−0.124 + 0.992i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + 2-s + (−0.661 − 0.749i)3-s + 4-s + (−0.733 + 0.680i)5-s + (−0.661 − 0.749i)6-s + 8-s + (−0.124 + 0.992i)9-s + (−0.733 + 0.680i)10-s + (0.456 + 0.889i)11-s + (−0.661 − 0.749i)12-s + (0.969 + 0.246i)13-s + (0.995 + 0.0995i)15-s + 16-s + (0.939 + 0.342i)17-s + (−0.124 + 0.992i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.933 - 0.358i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.933 - 0.358i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (4871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.933 - 0.358i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(3.119494843 - 0.5777089865i\)
\(L(\frac12)\) \(\approx\) \(3.119494843 - 0.5777089865i\)
\(L(1)\) \(\approx\) \(1.699671941 - 0.1564583089i\)
\(L(1)\) \(\approx\) \(1.699671941 - 0.1564583089i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + T \)
3 \( 1 + (-0.661 - 0.749i)T \)
5 \( 1 + (-0.733 + 0.680i)T \)
11 \( 1 + (0.456 + 0.889i)T \)
13 \( 1 + (0.969 + 0.246i)T \)
17 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.0249 - 0.999i)T \)
29 \( 1 + (-0.980 - 0.198i)T \)
31 \( 1 + (0.998 - 0.0498i)T \)
37 \( 1 + (0.921 - 0.388i)T \)
41 \( 1 + (-0.921 + 0.388i)T \)
43 \( 1 + (-0.173 - 0.984i)T \)
47 \( 1 + (0.988 + 0.149i)T \)
53 \( 1 + (0.661 - 0.749i)T \)
59 \( 1 + (0.698 + 0.715i)T \)
61 \( 1 + (-0.365 + 0.930i)T \)
67 \( 1 + (-0.698 + 0.715i)T \)
71 \( 1 + (-0.853 - 0.521i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (-0.661 - 0.749i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (0.955 + 0.294i)T \)
97 \( 1 + (0.995 + 0.0995i)T \)
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

−17.227614965728875634879242924508, −16.79886796782717363331788104897, −16.2105869400661199227431669047, −15.80344786746991783743300391465, −15.210584211469562676392134289333, −14.47509243578400510312937584954, −13.75728370354674280110618675405, −13.08885682615846255265094126195, −12.29816816211361786304698678932, −11.66758051755667809436584951729, −11.44691494072506573193562348493, −10.67585652047181770329814866799, −9.89035132953241857337218699384, −9.12737324789737872231163232836, −8.22322518974919110639047360227, −7.65795819466628920753520972045, −6.683837590440055675137979170810, −5.82077290160822228411801839173, −5.55871593103100331120078353365, −4.8042523079003474122414027352, −3.89950731269391321812115226593, −3.61583434419912087000512883353, −2.99114247704101375106871946273, −1.353590241331724585344597192, −0.95460102760328319659437203247, 0.76926116978189037007097927206, 1.648328971305079958428994670717, 2.46684061228517241761181857691, 3.20172976799675141543507623322, 4.11815439549729569286149150700, 4.56188595397903065589514711320, 5.5722845840785628081861486189, 6.14604109104310043101133995173, 6.8549321065574661395658351712, 7.26255032262890441275007172806, 7.90364052106512768707043806470, 8.75645189278023109034767827377, 10.09596392408246073487121224425, 10.5862270107020382406897930547, 11.32083326076147831832778333596, 11.9055523651898386757387910475, 12.14095557184525030414092056358, 13.120119164726641783508979383212, 13.51758104066167730954214005446, 14.37981424814519966065941910848, 14.90436816399526649652468083868, 15.52944818744138908633027464804, 16.32154829080714898136432417961, 16.7558957453313707369807142130, 17.624354895842372263861165444973

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