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Properties

Label 1-6223-6223.614-r0-0-0
Degree $1$
Conductor $6223$
Sign $0.899 - 0.437i$
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  + (0.365 − 0.930i)2-s + (0.456 − 0.889i)3-s + (−0.733 − 0.680i)4-s + (−0.5 + 0.866i)5-s + (−0.661 − 0.749i)6-s + (−0.900 + 0.433i)8-s + (−0.583 − 0.811i)9-s + (0.623 + 0.781i)10-s + (0.270 − 0.962i)11-s + (−0.939 + 0.342i)12-s + (0.797 − 0.603i)13-s + (0.542 + 0.840i)15-s + (0.0747 + 0.997i)16-s + (0.797 + 0.603i)17-s + (−0.969 + 0.246i)18-s − 19-s + ⋯
L(s)  = 1  + (0.365 − 0.930i)2-s + (0.456 − 0.889i)3-s + (−0.733 − 0.680i)4-s + (−0.5 + 0.866i)5-s + (−0.661 − 0.749i)6-s + (−0.900 + 0.433i)8-s + (−0.583 − 0.811i)9-s + (0.623 + 0.781i)10-s + (0.270 − 0.962i)11-s + (−0.939 + 0.342i)12-s + (0.797 − 0.603i)13-s + (0.542 + 0.840i)15-s + (0.0747 + 0.997i)16-s + (0.797 + 0.603i)17-s + (−0.969 + 0.246i)18-s − 19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.899 - 0.437i$
Analytic conductor: \(28.8994\)
Root analytic conductor: \(28.8994\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (614, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (0:\ ),\ 0.899 - 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.367775157 - 0.3148681856i\)
\(L(\frac12)\) \(\approx\) \(1.367775157 - 0.3148681856i\)
\(L(1)\) \(\approx\) \(0.9248212829 - 0.6727653019i\)
\(L(1)\) \(\approx\) \(0.9248212829 - 0.6727653019i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 \)
127 \( 1 \)
good2 \( 1 + (0.365 - 0.930i)T \)
3 \( 1 + (0.456 - 0.889i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
11 \( 1 + (0.270 - 0.962i)T \)
13 \( 1 + (0.797 - 0.603i)T \)
17 \( 1 + (0.797 + 0.603i)T \)
19 \( 1 - T \)
23 \( 1 + (-0.173 + 0.984i)T \)
29 \( 1 + (0.411 - 0.911i)T \)
31 \( 1 + (0.411 + 0.911i)T \)
37 \( 1 + (0.456 + 0.889i)T \)
41 \( 1 + (-0.173 - 0.984i)T \)
43 \( 1 + (-0.270 + 0.962i)T \)
47 \( 1 + (-0.0747 + 0.997i)T \)
53 \( 1 + (0.939 + 0.342i)T \)
59 \( 1 + (-0.411 + 0.911i)T \)
61 \( 1 + (-0.0747 + 0.997i)T \)
67 \( 1 + (-0.878 + 0.478i)T \)
71 \( 1 + (-0.939 + 0.342i)T \)
73 \( 1 + (0.988 + 0.149i)T \)
79 \( 1 + (-0.124 - 0.992i)T \)
83 \( 1 + (-0.411 + 0.911i)T \)
89 \( 1 + (-0.900 + 0.433i)T \)
97 \( 1 + (-0.318 + 0.947i)T \)
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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

−17.313116378546836937312746152005, −16.68707768814066614263430454557, −16.40841296148548552604068117437, −15.7563893688982884547174596773, −15.08811386402434228167974158224, −14.65689114576044593742243317923, −13.94432132067143856283825228793, −13.27803146109416374323761883841, −12.54197998649443241827892640490, −11.979243902070128059622716244710, −11.18615069907999403510635519079, −10.18486850801512663764383822521, −9.50666479504246046773109149490, −8.90639187391606548875061293026, −8.39762835861058476210225304575, −7.81016085351191966001975767170, −6.98099436249831437689257826793, −6.20726899580148896888687601925, −5.29264930892710991581932455694, −4.751504303848997187488803513549, −4.13651836052700273220984153267, −3.74715143986165730959959024045, −2.74227592670617384583316584331, −1.702496447168375892482985153692, −0.32649452062500714598544660545, 0.9678836303531089655217222377, 1.47144648602208457315998374951, 2.620798990938139561176548665686, 3.02689390387919805556240290235, 3.68905019363619428915414228810, 4.26222708983271129649658061645, 5.70108908062607057786912979269, 6.026075950731760339980902344128, 6.72186388794027416432356537048, 7.76968798232371416085757451388, 8.315339360500325886231509082643, 8.812895426485457958405680143203, 9.849946870806978212090664832534, 10.53564620800825138066874234028, 11.110941299144973162167294403031, 11.78650428496702046143447634660, 12.246034201987016611214515629039, 13.08054511549001770906883637734, 13.68758469740215871755411247116, 14.0408222553134551195828657857, 14.991054058040411365960702307632, 15.149646348118249358981401280663, 16.21450550802511745483643662541, 17.32891940004800745698480141272, 17.83269041265565133186516139035

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