↑

Properties

Label 1-6223-6223.2379-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.880 - 0.473i$
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  + (−0.222 − 0.974i)2-s + (−0.939 + 0.342i)3-s + (−0.900 + 0.433i)4-s + (−0.988 − 0.149i)5-s + (0.542 + 0.840i)6-s + (0.623 + 0.781i)8-s + (0.766 − 0.642i)9-s + (0.0747 + 0.997i)10-s + (0.173 + 0.984i)11-s + (0.698 − 0.715i)12-s + (−0.878 + 0.478i)13-s + (0.980 − 0.198i)15-s + (0.623 − 0.781i)16-s + (0.969 − 0.246i)17-s + (−0.797 − 0.603i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.939 + 0.342i)3-s + (−0.900 + 0.433i)4-s + (−0.988 − 0.149i)5-s + (0.542 + 0.840i)6-s + (0.623 + 0.781i)8-s + (0.766 − 0.642i)9-s + (0.0747 + 0.997i)10-s + (0.173 + 0.984i)11-s + (0.698 − 0.715i)12-s + (−0.878 + 0.478i)13-s + (0.980 − 0.198i)15-s + (0.623 − 0.781i)16-s + (0.969 − 0.246i)17-s + (−0.797 − 0.603i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.880 - 0.473i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.880 - 0.473i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (2379, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.880 - 0.473i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1255589347 - 0.4983049339i\)
\(L(\frac12)\) \(\approx\) \(0.1255589347 - 0.4983049339i\)
\(L(1)\) \(\approx\) \(0.4975038211 - 0.2224188216i\)
\(L(1)\) \(\approx\) \(0.4975038211 - 0.2224188216i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.222 - 0.974i)T \)
3 \( 1 + (-0.939 + 0.342i)T \)
5 \( 1 + (-0.988 - 0.149i)T \)
11 \( 1 + (0.173 + 0.984i)T \)
13 \( 1 + (-0.878 + 0.478i)T \)
17 \( 1 + (0.969 - 0.246i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (0.583 - 0.811i)T \)
29 \( 1 + (-0.921 + 0.388i)T \)
31 \( 1 + (-0.542 - 0.840i)T \)
37 \( 1 + (0.542 - 0.840i)T \)
41 \( 1 + (0.853 - 0.521i)T \)
43 \( 1 + (0.853 - 0.521i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.698 - 0.715i)T \)
59 \( 1 + (-0.411 + 0.911i)T \)
61 \( 1 + (0.733 - 0.680i)T \)
67 \( 1 + (0.969 + 0.246i)T \)
71 \( 1 + (-0.411 - 0.911i)T \)
73 \( 1 + (0.733 + 0.680i)T \)
79 \( 1 + (-0.318 + 0.947i)T \)
83 \( 1 + (0.173 + 0.984i)T \)
89 \( 1 + (-0.5 + 0.866i)T \)
97 \( 1 + (-0.411 - 0.911i)T \)
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

−17.77949426247825975820519742895, −17.12291850003603906070157787594, −16.61451128391560788957157835339, −16.073362540347399717692755110557, −15.57374978457831420580809005466, −14.63361508440721778019628256903, −14.32978757356007318736451380858, −13.290291853702452735956676764715, −12.66853773102646037220156027181, −12.097429180239797309812826130867, −11.27332657730313222844600501056, −10.77335801665500235953618061823, −9.919820081238387722566542369531, −9.30660191025685840651232002304, −8.22159224943875014954507273283, −7.661710601499425240536650520394, −7.43863323541419083162476364034, −6.46770088155927883447939845561, −5.81493564664574022111084722886, −5.30122829551204445338152389296, −4.552597807555728711500753453379, −3.72071192620596313380303502946, −3.0076829846056587307729213212, −1.39369982428271508027171966738, −0.8303379648821617962020260573, 0.27119649143583458695841760017, 0.96985355596118178429943853931, 2.0350717689893078849694898965, 2.87495571290077017126305609399, 3.948000485152806981929863706693, 4.18023232142879357315874121689, 5.08665381690151277248281459834, 5.419010201797260095930896996681, 6.930593230226259875567270217191, 7.263058490334193783878049979855, 7.99782203273112571186804868907, 9.1978838122120168931543140354, 9.41607672160246855161139428056, 10.17503030677075217315637583950, 11.07672421807054606364751650531, 11.29550622292513959472481244687, 12.135887555549647809874895772806, 12.49435364324806162639806389443, 12.94649802352823798444522652960, 14.1803948218551502324369032107, 14.80883220433437844787978630446, 15.4092597993632444998654607928, 16.41970456595364925122854944024, 16.717904039494042445727625202388, 17.4003370779018981551979228442

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