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Properties

Label 1-6223-6223.8-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.453 - 0.891i$
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  + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (−0.222 + 0.974i)6-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)11-s + 12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + (0.623 − 0.781i)18-s + 19-s + ⋯
L(s)  = 1  + (−0.222 − 0.974i)2-s + (−0.900 − 0.433i)3-s + (−0.900 + 0.433i)4-s + 5-s + (−0.222 + 0.974i)6-s + (0.623 + 0.781i)8-s + (0.623 + 0.781i)9-s + (−0.222 − 0.974i)10-s + (0.623 − 0.781i)11-s + 12-s + (−0.900 + 0.433i)13-s + (−0.900 − 0.433i)15-s + (0.623 − 0.781i)16-s + (−0.900 − 0.433i)17-s + (0.623 − 0.781i)18-s + 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.453 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.453 - 0.891i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (8, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.453 - 0.891i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.231606844 - 0.7552060262i\)
\(L(\frac12)\) \(\approx\) \(1.231606844 - 0.7552060262i\)
\(L(1)\) \(\approx\) \(0.7698452669 - 0.4378280232i\)
\(L(1)\) \(\approx\) \(0.7698452669 - 0.4378280232i\)

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 \( 1 + (-0.900 - 0.433i)T \)
5 \( 1 + T \)
11 \( 1 + (0.623 - 0.781i)T \)
13 \( 1 + (-0.900 + 0.433i)T \)
17 \( 1 + (-0.900 - 0.433i)T \)
19 \( 1 + T \)
23 \( 1 + T \)
29 \( 1 + (0.623 + 0.781i)T \)
31 \( 1 + (0.623 - 0.781i)T \)
37 \( 1 + (-0.900 + 0.433i)T \)
41 \( 1 + T \)
43 \( 1 + (0.623 - 0.781i)T \)
47 \( 1 + (0.623 + 0.781i)T \)
53 \( 1 + T \)
59 \( 1 + (0.623 + 0.781i)T \)
61 \( 1 + (0.623 + 0.781i)T \)
67 \( 1 + (-0.222 - 0.974i)T \)
71 \( 1 + T \)
73 \( 1 + (-0.222 + 0.974i)T \)
79 \( 1 + (-0.900 + 0.433i)T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
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

−17.50476944982869212472789067295, −17.394680145895157522864657843651, −16.55072103483960359233896911789, −15.85998031887130785725828716093, −15.26735278754284317887979034351, −14.64541934979125362527649173289, −14.05773634551111711719227014567, −13.19172265016164602860503048150, −12.63511536001489399598497718395, −11.94683324907481666222440299200, −10.95831413770289352362757028849, −10.22509731660884611930575208182, −9.82475240493698163277965055440, −9.21631592663237928449999010084, −8.61273263849507657162955636579, −7.374583855932558397341983146934, −6.96278360132332287436289878501, −6.30459379828925442988837426781, −5.664868523242676386383972574089, −4.94507129357114007870547431062, −4.6116466297646318693672166986, −3.6470103955073944691223873349, −2.455882565022271113853124396248, −1.41953767590306568093860878456, −0.63496914989778254304928350204, 0.86793379965353713649072582212, 1.19100661298492988863866922469, 2.34102692366515227199959362244, 2.65421150622450526385700786434, 3.842636865232254981426261775577, 4.73276763649144867740878900181, 5.236265572869246677287744730554, 5.93772159631700225693576385245, 6.87023836086628412646487390, 7.297705037713622001492242794129, 8.451982721915545641090429202321, 9.149646219072833453302487180638, 9.63305783319591269809965028589, 10.4277361604488180321411398194, 10.98145746590882546102821875381, 11.61151011493586904974162742658, 12.172531921329826332243318685485, 12.80281377298795187595767227115, 13.70286605550850522320892953698, 13.75761287490422131174073729011, 14.64276562886396169094140950904, 15.836945299363083845477537148, 16.63871105850907158936666651264, 17.069508953432599184283990660011, 17.64375965866643889209845482954

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