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Properties

Label 1-6223-6223.1762-r0-0-0
Degree $1$
Conductor $6223$
Sign $-0.998 + 0.0457i$
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  + (−0.5 − 0.866i)2-s + (0.826 − 0.563i)3-s + (−0.5 + 0.866i)4-s + (0.365 − 0.930i)5-s + (−0.900 − 0.433i)6-s + 8-s + (0.365 − 0.930i)9-s + (−0.988 + 0.149i)10-s + (0.365 + 0.930i)11-s + (0.0747 + 0.997i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.988 + 0.149i)18-s + (0.5 + 0.866i)19-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)2-s + (0.826 − 0.563i)3-s + (−0.5 + 0.866i)4-s + (0.365 − 0.930i)5-s + (−0.900 − 0.433i)6-s + 8-s + (0.365 − 0.930i)9-s + (−0.988 + 0.149i)10-s + (0.365 + 0.930i)11-s + (0.0747 + 0.997i)12-s + (0.222 − 0.974i)13-s + (−0.222 − 0.974i)15-s + (−0.5 − 0.866i)16-s + (0.5 − 0.866i)17-s + (−0.988 + 0.149i)18-s + (0.5 + 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.998 + 0.0457i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $-0.998 + 0.0457i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (1762, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ -0.998 + 0.0457i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.04789540201 - 2.090674084i\)
\(L(\frac12)\) \(\approx\) \(-0.04789540201 - 2.090674084i\)
\(L(1)\) \(\approx\) \(0.8028740207 - 0.9098440620i\)
\(L(1)\) \(\approx\) \(0.8028740207 - 0.9098440620i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (0.826 - 0.563i)T \)
5 \( 1 + (0.365 - 0.930i)T \)
11 \( 1 + (0.365 + 0.930i)T \)
13 \( 1 + (0.222 - 0.974i)T \)
17 \( 1 + (0.5 - 0.866i)T \)
19 \( 1 + (0.5 + 0.866i)T \)
23 \( 1 + (-0.826 - 0.563i)T \)
29 \( 1 + (0.900 - 0.433i)T \)
31 \( 1 + (0.988 - 0.149i)T \)
37 \( 1 + (0.365 - 0.930i)T \)
41 \( 1 + (-0.623 - 0.781i)T \)
43 \( 1 - T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.0747 + 0.997i)T \)
59 \( 1 + (0.955 + 0.294i)T \)
61 \( 1 + (-0.826 - 0.563i)T \)
67 \( 1 + (0.733 + 0.680i)T \)
71 \( 1 + (-0.900 + 0.433i)T \)
73 \( 1 + (0.5 - 0.866i)T \)
79 \( 1 + (0.0747 + 0.997i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.365 - 0.930i)T \)
97 \( 1 + (-0.222 - 0.974i)T \)
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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

−17.95005399856944656214114005441, −17.288888952708333838258289424103, −16.49819580509385814903072983394, −16.050469611687359727609206593921, −15.349885094796245201402299748387, −14.72412988829982248347915223249, −14.25490707365566711184842543608, −13.574212944619268123415680623018, −13.41933827661176481299008157177, −11.841865436714874146396956140215, −11.14745430782427619848510481043, −10.485563682318210658177320884108, −9.79476860740273980082371975964, −9.45086440722422828487784685039, −8.46696310820598503368334576527, −8.212238075758758526436427090554, −7.296581849991730914962205574808, −6.5324908435162413464070978228, −6.14460116120221411995274935965, −5.15892527113883720587374771167, −4.42916965269276700996391769656, −3.55439844848518678854535340298, −2.97465630038972020763059323516, −1.90706869168022713852044563487, −1.22818852013580485318510026641, 0.589045876633342560588272144537, 1.192653892692832716283581782, 1.992245294839579323173403841073, 2.566391895657342445762685548536, 3.447922263301467698064216726541, 4.13564131110298683128091037912, 4.88807504631375075761066602591, 5.78401547869791293196805642536, 6.78809561467853130725591292359, 7.6297487517044493185083181011, 8.13220438541953682735294145515, 8.64942548901380182931977762440, 9.44752274568691053524523511568, 9.95693414567685234573450708378, 10.34776207067196253115198545913, 11.82420944751383148960954584821, 12.06332639616258329570075143431, 12.599805771369903889110014456201, 13.303885409204199261120086767439, 13.85981239399818803677691875972, 14.40759456855221744169728718293, 15.44808825620754476326325211567, 16.08728652769819235950985557364, 16.910367766327643428727183997809, 17.5322154192150926299351151077

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