Properties

Label 1-456-456.53-r0-0-0
Degree $1$
Conductor $456$
Sign $-0.973 - 0.226i$
Analytic cond. $2.11765$
Root an. cond. $2.11765$
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  + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.766 − 0.642i)17-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 + 0.984i)43-s + (−0.766 + 0.642i)47-s + (−0.5 + 0.866i)49-s + ⋯
L(s)  = 1  + (0.173 − 0.984i)5-s + (−0.5 − 0.866i)7-s + (−0.5 + 0.866i)11-s + (−0.939 − 0.342i)13-s + (−0.766 − 0.642i)17-s + (−0.173 − 0.984i)23-s + (−0.939 − 0.342i)25-s + (−0.766 + 0.642i)29-s + (0.5 + 0.866i)31-s + (−0.939 + 0.342i)35-s + 37-s + (−0.939 + 0.342i)41-s + (−0.173 + 0.984i)43-s + (−0.766 + 0.642i)47-s + (−0.5 + 0.866i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 456 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.973 - 0.226i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(456\)    =    \(2^{3} \cdot 3 \cdot 19\)
Sign: $-0.973 - 0.226i$
Analytic conductor: \(2.11765\)
Root analytic conductor: \(2.11765\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{456} (53, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 456,\ (0:\ ),\ -0.973 - 0.226i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.06011660039 - 0.5235260987i\)
\(L(\frac12)\) \(\approx\) \(0.06011660039 - 0.5235260987i\)
\(L(1)\) \(\approx\) \(0.7188510925 - 0.2774825934i\)
\(L(1)\) \(\approx\) \(0.7188510925 - 0.2774825934i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
19 \( 1 \)
good5 \( 1 + (0.173 - 0.984i)T \)
7 \( 1 + (-0.5 - 0.866i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
13 \( 1 + (-0.939 - 0.342i)T \)
17 \( 1 + (-0.766 - 0.642i)T \)
23 \( 1 + (-0.173 - 0.984i)T \)
29 \( 1 + (-0.766 + 0.642i)T \)
31 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 + T \)
41 \( 1 + (-0.939 + 0.342i)T \)
43 \( 1 + (-0.173 + 0.984i)T \)
47 \( 1 + (-0.766 + 0.642i)T \)
53 \( 1 + (-0.173 - 0.984i)T \)
59 \( 1 + (-0.766 - 0.642i)T \)
61 \( 1 + (-0.173 - 0.984i)T \)
67 \( 1 + (0.766 - 0.642i)T \)
71 \( 1 + (0.173 - 0.984i)T \)
73 \( 1 + (-0.939 + 0.342i)T \)
79 \( 1 + (0.939 - 0.342i)T \)
83 \( 1 + (-0.5 - 0.866i)T \)
89 \( 1 + (-0.939 - 0.342i)T \)
97 \( 1 + (-0.766 - 0.642i)T \)
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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

−24.36399043333702703516862393025, −23.50878325206296156365455845713, −22.36216455576760940491528157990, −21.90180417588376644767086206658, −21.267781378689205470848682081662, −19.85363641782896513423915706051, −19.03641863222128540174747612923, −18.557642646229528254792325489930, −17.535877009398847330287751272424, −16.60917690308417223766593067477, −15.37523977246358529155886387709, −15.07639449502543606017547974039, −13.816922190295653379275824077519, −13.13055851957158110090003135439, −11.89994716598156020865809714132, −11.19654358218078494626883836986, −10.126767845825832800745828938206, −9.3525297369149591027603497982, −8.23002629702341107729897444471, −7.17229723014945255085957492396, −6.17092247177696927904129434895, −5.47974098856205841769742922769, −3.94667805157283801104370288732, −2.83793293464034462510844161606, −2.0953733309391712196740967881, 0.27049707862006937136302048923, 1.77706143296603940165383149807, 3.04275298411386329546709464410, 4.54641309493092923267566084957, 4.92688240789689399178747253223, 6.40928955191955627289837846876, 7.35165937897044632679384562480, 8.2725590920312038389074171949, 9.51271210384495400616705295777, 10.00576525233710624489135900696, 11.15831741941474141636955047769, 12.48094947012380367032690854446, 12.86732754672045855823998402499, 13.82520989372269019249237622755, 14.879797043189294875287129436203, 15.950105548263188261348500767186, 16.63427495658278000681222818171, 17.45240127936832029166681520129, 18.2294366449251748811539946918, 19.61982458719083539289452333764, 20.16632005783417451825704539018, 20.73323181499762306419826955170, 21.90976070056646615448715454020, 22.779100656332219885603806978344, 23.554719997864379499733744522062

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