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Properties

Label 1-6223-6223.2295-r1-0-0
Degree $1$
Conductor $6223$
Sign $0.807 + 0.590i$
Analytic cond. $668.754$
Root an. cond. $668.754$
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 + (−0.698 + 0.715i)3-s + (0.623 − 0.781i)4-s + (−0.0747 − 0.997i)5-s + (0.318 − 0.947i)6-s + (−0.222 + 0.974i)8-s + (−0.0249 − 0.999i)9-s + (0.5 + 0.866i)10-s + (−0.661 + 0.749i)11-s + (0.124 + 0.992i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.222 − 0.974i)16-s + (−0.980 − 0.198i)17-s + (0.456 + 0.889i)18-s + (0.5 − 0.866i)19-s + ⋯
L(s)  = 1  + (−0.900 + 0.433i)2-s + (−0.698 + 0.715i)3-s + (0.623 − 0.781i)4-s + (−0.0747 − 0.997i)5-s + (0.318 − 0.947i)6-s + (−0.222 + 0.974i)8-s + (−0.0249 − 0.999i)9-s + (0.5 + 0.866i)10-s + (−0.661 + 0.749i)11-s + (0.124 + 0.992i)12-s + (−0.173 + 0.984i)13-s + (0.766 + 0.642i)15-s + (−0.222 − 0.974i)16-s + (−0.980 − 0.198i)17-s + (0.456 + 0.889i)18-s + (0.5 − 0.866i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6223 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.807 + 0.590i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(6223\)    =    \(7^{2} \cdot 127\)
Sign: $0.807 + 0.590i$
Analytic conductor: \(668.754\)
Root analytic conductor: \(668.754\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{6223} (2295, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 6223,\ (1:\ ),\ 0.807 + 0.590i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2399463550 + 0.07835127607i\)
\(L(\frac12)\) \(\approx\) \(0.2399463550 + 0.07835127607i\)
\(L(1)\) \(\approx\) \(0.4127558783 + 0.1102000229i\)
\(L(1)\) \(\approx\) \(0.4127558783 + 0.1102000229i\)

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 \( 1 + (-0.698 + 0.715i)T \)
5 \( 1 + (-0.0747 - 0.997i)T \)
11 \( 1 + (-0.661 + 0.749i)T \)
13 \( 1 + (-0.173 + 0.984i)T \)
17 \( 1 + (-0.980 - 0.198i)T \)
19 \( 1 + (0.5 - 0.866i)T \)
23 \( 1 + (-0.998 + 0.0498i)T \)
29 \( 1 + (0.173 + 0.984i)T \)
31 \( 1 + (0.124 + 0.992i)T \)
37 \( 1 + (0.995 - 0.0995i)T \)
41 \( 1 + (-0.698 - 0.715i)T \)
43 \( 1 + (-0.124 + 0.992i)T \)
47 \( 1 + (-0.0747 - 0.997i)T \)
53 \( 1 + (-0.124 + 0.992i)T \)
59 \( 1 + (-0.980 - 0.198i)T \)
61 \( 1 + (0.5 + 0.866i)T \)
67 \( 1 + (-0.411 - 0.911i)T \)
71 \( 1 + (0.456 - 0.889i)T \)
73 \( 1 + (-0.826 + 0.563i)T \)
79 \( 1 + (-0.853 - 0.521i)T \)
83 \( 1 + (-0.878 - 0.478i)T \)
89 \( 1 + (-0.365 - 0.930i)T \)
97 \( 1 + (0.969 + 0.246i)T \)
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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

−17.552329300322907441136216184714, −17.103250172479456554181681348, −16.168253752948413347218424277161, −15.73282701608596389456257226477, −14.99241320256070745566022552493, −13.9547669088317493623244768450, −13.30380840852271467217466996064, −12.764180760821423776184828189004, −11.88622034949574848757710451016, −11.37554716467157592540307444449, −10.924659332706319419777978369395, −10.10292892385356200481526761204, −9.908144774084345706695618198430, −8.532898473622538033242550800364, −7.810147624613964515498666604227, −7.74429405078785602820409628561, −6.68546412094283538175582213420, −6.11916778768589300328113320647, −5.60711376960988572300571643102, −4.323080190908731773288660098838, −3.43825455636376594970901543659, −2.59790347178088520909922991013, −2.19303194216209716494057583270, −1.15295098371607882215751579544, −0.22271021626314507823822963564, 0.20485326314315027710816143363, 1.26718435913924771375820266657, 1.975663007765994981870383092763, 3.006259044169541616598890512369, 4.33830384525769300215990818540, 4.71595058964984667740790794621, 5.3203827270130593712810070828, 6.11997036438894631697018239450, 6.882821411898983486653391358023, 7.44546786197933042340837849002, 8.44934556540944762234354544802, 9.04824299672887055938498572364, 9.46351183303652750310279504235, 10.17626535871180151642785693584, 10.79203600306569706318453485331, 11.63479538545076158073059712733, 11.9739775276419700754738639520, 12.83847353173823526812723901529, 13.72538544957745397687527490685, 14.53454157766819760275824216144, 15.35848809354762274549720334974, 15.86980566541294098059553485392, 16.182269361789642158916183738295, 16.91136754806279026059464523760, 17.445332941096757600709851580928

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