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Properties

Label 1-6223-6223.1049-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.702 + 0.711i$
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  + (−0.900 − 0.433i)2-s + (−0.988 + 0.149i)3-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (0.955 + 0.294i)6-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)9-s + (0.623 + 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.733 − 0.680i)12-s + (0.733 + 0.680i)13-s + (0.955 + 0.294i)15-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (−0.988 − 0.149i)18-s − 19-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)2-s + (−0.988 + 0.149i)3-s + (0.623 + 0.781i)4-s + (−0.900 − 0.433i)5-s + (0.955 + 0.294i)6-s + (−0.222 − 0.974i)8-s + (0.955 − 0.294i)9-s + (0.623 + 0.781i)10-s + (−0.733 + 0.680i)11-s + (−0.733 − 0.680i)12-s + (0.733 + 0.680i)13-s + (0.955 + 0.294i)15-s + (−0.222 + 0.974i)16-s + (0.988 + 0.149i)17-s + (−0.988 − 0.149i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.702 + 0.711i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.702 + 0.711i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1049, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.702 + 0.711i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2677306824 + 0.1118637197i\)
\(L(\frac12)\) \(\approx\) \(0.2677306824 + 0.1118637197i\)
\(L(1)\) \(\approx\) \(0.3906336769 - 0.04662597922i\)
\(L(1)\) \(\approx\) \(0.3906336769 - 0.04662597922i\)

Euler product

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

−17.737005794596173464753836677775, −16.70576494382692091086109867765, −16.40407950840172895691939486185, −15.801460632980614451245163472115, −15.28015054624720207441056801454, −14.563131441086127512470541074227, −13.70817910995785819069179547839, −12.76419233624344993020948435467, −12.17795790306403922821790812332, −11.30866783457176419796113847782, −10.98715694983507163295428390733, −10.37231579352916386662746117398, −9.86626072108477466360624953832, −8.636736982324250389499332749973, −8.151764755128050018705016398351, −7.57382194899014128420456856657, −6.88323140962481165435615440668, −6.21826151606918722516180368394, −5.53683381614124850185479402133, −5.021607737007077166569151344595, −3.817737320849515198820604275188, −3.16057700003013517508169745684, −2.02511846133065870727016795342, −1.12221310365428441537718223152, −0.24080066767062649443384101128, 0.51224385235909782619867477720, 1.57582856665857607686396177187, 2.1496522310486143316607024725, 3.54446739020494492442429032888, 3.94874552766253612859674429411, 4.71265182735969867615860768351, 5.608051546534519040219470874250, 6.45513951335718466270375990716, 7.095883356435835175454415153688, 7.874515174629003507783220391329, 8.30195929321730133100851644763, 9.2214708013067422293989504169, 9.92978812529875601153279070307, 10.49879794830141170102599445768, 11.13359282863519794415612203548, 11.802637195712864328130752938008, 12.166457741809232943032744271986, 12.86838018589921613991867990193, 13.42232572910201473623722509250, 14.92333911725831977763035643770, 15.30895949375707332264700056124, 16.1216130622296676448984293069, 16.51585272774188209115873321228, 16.98664458878477059686414227551, 17.76965157347403763325125673978

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