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Properties

Label 1-6223-6223.5297-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.229 - 0.973i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.955 − 0.294i)2-s + (0.980 − 0.198i)3-s + (0.826 − 0.563i)4-s + (0.365 + 0.930i)5-s + (0.878 − 0.478i)6-s + (0.623 − 0.781i)8-s + (0.921 − 0.388i)9-s + (0.623 + 0.781i)10-s + (−0.998 − 0.0498i)11-s + (0.698 − 0.715i)12-s + (0.797 − 0.603i)13-s + (0.542 + 0.840i)15-s + (0.365 − 0.930i)16-s + (0.411 − 0.911i)17-s + (0.766 − 0.642i)18-s − 19-s + ⋯
L(s)  = 1  + (0.955 − 0.294i)2-s + (0.980 − 0.198i)3-s + (0.826 − 0.563i)4-s + (0.365 + 0.930i)5-s + (0.878 − 0.478i)6-s + (0.623 − 0.781i)8-s + (0.921 − 0.388i)9-s + (0.623 + 0.781i)10-s + (−0.998 − 0.0498i)11-s + (0.698 − 0.715i)12-s + (0.797 − 0.603i)13-s + (0.542 + 0.840i)15-s + (0.365 − 0.930i)16-s + (0.411 − 0.911i)17-s + (0.766 − 0.642i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.229 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.229 - 0.973i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (5297, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.229 - 0.973i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(4.441575008 - 3.515109174i\)
\(L(\frac12)\) \(\approx\) \(4.441575008 - 3.515109174i\)
\(L(1)\) \(\approx\) \(2.651126636 - 0.8704289021i\)
\(L(1)\) \(\approx\) \(2.651126636 - 0.8704289021i\)

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 \( 1 + (0.980 - 0.198i)T \)
5 \( 1 + (0.365 + 0.930i)T \)
11 \( 1 + (-0.998 - 0.0498i)T \)
13 \( 1 + (0.797 - 0.603i)T \)
17 \( 1 + (0.411 - 0.911i)T \)
19 \( 1 - T \)
23 \( 1 + (0.583 - 0.811i)T \)
29 \( 1 + (0.411 - 0.911i)T \)
31 \( 1 + (-0.766 - 0.642i)T \)
37 \( 1 + (-0.411 + 0.911i)T \)
41 \( 1 + (-0.878 - 0.478i)T \)
43 \( 1 + (0.661 - 0.749i)T \)
47 \( 1 + (0.733 + 0.680i)T \)
53 \( 1 + (-0.698 - 0.715i)T \)
59 \( 1 + (0.980 + 0.198i)T \)
61 \( 1 + (-0.0747 + 0.997i)T \)
67 \( 1 + (-0.173 + 0.984i)T \)
71 \( 1 + (0.995 + 0.0995i)T \)
73 \( 1 + (0.733 - 0.680i)T \)
79 \( 1 + (-0.939 + 0.342i)T \)
83 \( 1 + (-0.797 - 0.603i)T \)
89 \( 1 + (-0.222 - 0.974i)T \)
97 \( 1 + (-0.939 + 0.342i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−17.64010924948613066596731775630, −16.825074194658725426779529834207, −16.32569739577785178215171683836, −15.65430741310130593253009511791, −15.249304013958918511281801160912, −14.36532838204207553165017613819, −13.90343910819338664170877553153, −13.24628594039019639342630226458, −12.65591937201333576709045516077, −12.45284473828293233395055155627, −11.10911807516681840171413838142, −10.66426755326164043038264015739, −9.80252949651318452474244263283, −8.87602663947478258131876037224, −8.462587251796267413911232094657, −7.84511191594971698323545869210, −7.04401328190282749006357887873, −6.26244699981365387419763285134, −5.388427888254189596113173446044, −4.93086722298559565407634357475, −4.06159099295011877519644290734, −3.594316034543187679417627077699, −2.718509622060082169290859147889, −1.85807673853259707286959246651, −1.41274637369474750320265628218, 0.75233859005580661646553416020, 1.86682655366636166784484748660, 2.53811877495342975259270619435, 2.92565599685858004702910590713, 3.64987469304608012636539096295, 4.38772465544300126903561235164, 5.33130460461501984994178621741, 6.00257967049363599070728901838, 6.77032482261579490596055613917, 7.31384679823486669758717934415, 8.04426621893100328826996807297, 8.81854566371858806282103917827, 9.867294285534222176380205878513, 10.29945567361926658598890906775, 10.86528562926270581178738684899, 11.61757348217134581477478495922, 12.56746298712320525571861646246, 13.111635148621130442780375595381, 13.59260169936923661242771337702, 14.17015420661030468680309996015, 14.76830923597253111807914085793, 15.465652291061794242245882964551, 15.671154042298323127740799597720, 16.69592658048471239732608022065, 17.70659137620907317386615184389

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