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Properties

Label 1-3920-3920.267-r0-0-0
Degree $1$
Conductor $3920$
Sign $0.545 + 0.838i$
Analytic cond. $18.2044$
Root an. cond. $18.2044$
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)3-s + (0.623 + 0.781i)9-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)17-s − i·19-s + (−0.974 + 0.222i)23-s + (−0.222 − 0.974i)27-s + (−0.974 − 0.222i)29-s − 31-s + (0.433 + 0.900i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + ⋯
L(s)  = 1  + (−0.900 − 0.433i)3-s + (0.623 + 0.781i)9-s + (−0.781 − 0.623i)11-s + (−0.623 + 0.781i)13-s + (−0.974 − 0.222i)17-s − i·19-s + (−0.974 + 0.222i)23-s + (−0.222 − 0.974i)27-s + (−0.974 − 0.222i)29-s − 31-s + (0.433 + 0.900i)33-s + (0.222 − 0.974i)37-s + (0.900 − 0.433i)39-s + (0.900 + 0.433i)41-s + (0.900 − 0.433i)43-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3920 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.545 + 0.838i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3920\)    =    \(2^{4} \cdot 5 \cdot 7^{2}\)
Sign: $0.545 + 0.838i$
Analytic conductor: \(18.2044\)
Root analytic conductor: \(18.2044\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3920} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3920,\ (0:\ ),\ 0.545 + 0.838i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.3171918310 + 0.1721228383i\)
\(L(\frac12)\) \(\approx\) \(0.3171918310 + 0.1721228383i\)
\(L(1)\) \(\approx\) \(0.5917981571 - 0.1000154808i\)
\(L(1)\) \(\approx\) \(0.5917981571 - 0.1000154808i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 \)
7 \( 1 \)
good3 \( 1 + (-0.900 - 0.433i)T \)
11 \( 1 + (-0.781 - 0.623i)T \)
13 \( 1 + (-0.623 + 0.781i)T \)
17 \( 1 + (-0.974 - 0.222i)T \)
19 \( 1 - iT \)
23 \( 1 + (-0.974 + 0.222i)T \)
29 \( 1 + (-0.974 - 0.222i)T \)
31 \( 1 - T \)
37 \( 1 + (0.222 - 0.974i)T \)
41 \( 1 + (0.900 + 0.433i)T \)
43 \( 1 + (0.900 - 0.433i)T \)
47 \( 1 + (-0.781 - 0.623i)T \)
53 \( 1 + (-0.222 - 0.974i)T \)
59 \( 1 + (-0.433 - 0.900i)T \)
61 \( 1 + (0.974 + 0.222i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.222 - 0.974i)T \)
73 \( 1 + (-0.781 + 0.623i)T \)
79 \( 1 + T \)
83 \( 1 + (0.623 + 0.781i)T \)
89 \( 1 + (0.623 + 0.781i)T \)
97 \( 1 + iT \)
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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

−18.1473525075260653030898122170, −17.77896709658205265406468972780, −17.11061531878582350548804973526, −16.300797903334395548373371472694, −15.839303371611254760926336403616, −15.0121866477048077840448733552, −14.63760217991285232263280867885, −13.46112027175725863178916936376, −12.62741178559572923368716096687, −12.427933986010929024189976380494, −11.420019977302891619872982014027, −10.768783896027368124193192746855, −10.17374594942369680681721241057, −9.634748568243985946631864083100, −8.7440303274760466474662146467, −7.66693260845075373655155315259, −7.31107013745250136709568276738, −6.10246795690188156350343819673, −5.78017132226276032304152845248, −4.83111437650680574799961345261, −4.3163883741398024294846692454, −3.41398648094341204997967286120, −2.38224524692302314145738756271, −1.5146339540198305870938738086, −0.17956230473261832036486908001, 0.61291893548550402932835014770, 2.00372125744422164645570775633, 2.34107878454701638147455106308, 3.66963441918802036234637813626, 4.5320071716628816774473115073, 5.20148828718238647510096808397, 5.89965079672038317991424096876, 6.64867107982227500273699525794, 7.349423023817806709207038796639, 7.91782703790802256152765185931, 8.988636995682269307672401487691, 9.59184870938797744985107922612, 10.57231561463038709984523775729, 11.1810302883048061519493907531, 11.57398408676066881126788200968, 12.496316130146271426309804955944, 13.125552687153677298161147894002, 13.63444663083227979208930260668, 14.480703747097565369262981149902, 15.38169654685722030103289397440, 16.23156254550259432123602059221, 16.38547100288799283111982574554, 17.46771919119386426594835204263, 17.869583961045297468268534427283, 18.49237348931308593785048176248

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