↑

Properties

Label 1-6223-6223.3112-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.998 + 0.0605i$
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  + (0.0747 − 0.997i)2-s + (0.365 − 0.930i)3-s + (−0.988 − 0.149i)4-s + (−0.5 + 0.866i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)8-s + (−0.733 − 0.680i)9-s + (0.826 + 0.563i)10-s + (−0.733 + 0.680i)11-s + (−0.5 + 0.866i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.733 + 0.680i)18-s + (−0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (0.0747 − 0.997i)2-s + (0.365 − 0.930i)3-s + (−0.988 − 0.149i)4-s + (−0.5 + 0.866i)5-s + (−0.900 − 0.433i)6-s + (−0.222 + 0.974i)8-s + (−0.733 − 0.680i)9-s + (0.826 + 0.563i)10-s + (−0.733 + 0.680i)11-s + (−0.5 + 0.866i)12-s + (0.623 − 0.781i)13-s + (0.623 + 0.781i)15-s + (0.955 + 0.294i)16-s + (0.365 − 0.930i)17-s + (−0.733 + 0.680i)18-s + (−0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0605i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.998 + 0.0605i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (3112, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.998 + 0.0605i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.02346821006 - 0.7750528510i\)
\(L(\frac12)\) \(\approx\) \(0.02346821006 - 0.7750528510i\)
\(L(1)\) \(\approx\) \(0.6674901526 - 0.5169918717i\)
\(L(1)\) \(\approx\) \(0.6674901526 - 0.5169918717i\)

Euler product

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

−17.7670995991004417997728858458, −16.90421586295627353424571244125, −16.557000420756513858935314746114, −16.00045667998053212915498145497, −15.56493360559241346815134277715, −14.821843256056507697847835116642, −14.31098699552569965671997279241, −13.40181844833818150061858685448, −13.08545916490162719725660229174, −12.24387909867149894463463091048, −11.198571126689061710626521131230, −10.76515203899450062856440282206, −9.71755354057348133631590802919, −9.17110780920959003865347136143, −8.61887843964657442405579113367, −8.01882631939654174879762422740, −7.62021262305360503722193298558, −6.28240364770014650116254725245, −5.87102324035573764835996464165, −5.06136157795320870877627916630, −4.27030427975513476083154767572, −4.07085987925723600245619431270, −3.16855716639329910577005126425, −2.10220525984250100826638287821, −0.75509781612683073271897350992, 0.24381104057286772454136038748, 1.348914533355445270894917141724, 2.06502130206503535053871752081, 2.79906811006157143713462396514, 3.403120327292121944455872937929, 3.9121654219933752198127396974, 5.173938701116497363423098969071, 5.67337300681757768101202224052, 6.71692450870967141190554941822, 7.364062576396592447048125979401, 8.12046179171679824176663089812, 8.423829766721619883693556307660, 9.61755055086656519424314828241, 10.040580142157238071467612275402, 10.95171256811762562761192866834, 11.33307311764911980270049932337, 12.20721111771831815600205317727, 12.61680508167099929623954368834, 13.24573481148778616516940656858, 14.02137950985796944152581169442, 14.4419341708511942962847879299, 15.15250999213590828099315833709, 15.82765283777411402409257738331, 16.89451037756452124285526286100, 17.94781554394886160944458601956

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