Properties

Label 1-384-384.125-r1-0-0
Degree $1$
Conductor $384$
Sign $0.970 - 0.242i$
Analytic cond. $41.2665$
Root an. cond. $41.2665$
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  + (−0.831 − 0.555i)5-s + (0.923 − 0.382i)7-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (0.980 + 0.195i)29-s i·31-s + (−0.980 − 0.195i)35-s + (−0.555 + 0.831i)37-s + (−0.382 + 0.923i)41-s + (−0.195 − 0.980i)43-s + (−0.707 − 0.707i)47-s + ⋯
L(s)  = 1  + (−0.831 − 0.555i)5-s + (0.923 − 0.382i)7-s + (−0.980 + 0.195i)11-s + (−0.831 + 0.555i)13-s + (0.707 + 0.707i)17-s + (0.555 + 0.831i)19-s + (0.382 − 0.923i)23-s + (0.382 + 0.923i)25-s + (0.980 + 0.195i)29-s i·31-s + (−0.980 − 0.195i)35-s + (−0.555 + 0.831i)37-s + (−0.382 + 0.923i)41-s + (−0.195 − 0.980i)43-s + (−0.707 − 0.707i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.970 - 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(384\)    =    \(2^{7} \cdot 3\)
Sign: $0.970 - 0.242i$
Analytic conductor: \(41.2665\)
Root analytic conductor: \(41.2665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{384} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 384,\ (1:\ ),\ 0.970 - 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.616175849 - 0.1993362785i\)
\(L(\frac12)\) \(\approx\) \(1.616175849 - 0.1993362785i\)
\(L(1)\) \(\approx\) \(1.007081209 - 0.07829955382i\)
\(L(1)\) \(\approx\) \(1.007081209 - 0.07829955382i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (-0.831 - 0.555i)T \)
7 \( 1 + (0.923 - 0.382i)T \)
11 \( 1 + (-0.980 + 0.195i)T \)
13 \( 1 + (-0.831 + 0.555i)T \)
17 \( 1 + (0.707 + 0.707i)T \)
19 \( 1 + (0.555 + 0.831i)T \)
23 \( 1 + (0.382 - 0.923i)T \)
29 \( 1 + (0.980 + 0.195i)T \)
31 \( 1 - iT \)
37 \( 1 + (-0.555 + 0.831i)T \)
41 \( 1 + (-0.382 + 0.923i)T \)
43 \( 1 + (-0.195 - 0.980i)T \)
47 \( 1 + (-0.707 - 0.707i)T \)
53 \( 1 + (0.980 - 0.195i)T \)
59 \( 1 + (0.831 + 0.555i)T \)
61 \( 1 + (0.195 - 0.980i)T \)
67 \( 1 + (0.195 - 0.980i)T \)
71 \( 1 + (0.923 - 0.382i)T \)
73 \( 1 + (0.923 + 0.382i)T \)
79 \( 1 + (0.707 - 0.707i)T \)
83 \( 1 + (0.555 + 0.831i)T \)
89 \( 1 + (0.382 + 0.923i)T \)
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

−24.24329266852034720858947772780, −23.47175245827413748290675524930, −22.70301743417108861242276134303, −21.66234326735917831487015719910, −20.94440409026426838992874919977, −19.85816853572216182724465079408, −19.11285942090855803293315555133, −18.08717588070122381267689600571, −17.60227276077140389623144561327, −16.09917917283696331880377772628, −15.48768169487428655438527602864, −14.639862801552047063240801505705, −13.78337837867837260998297567975, −12.47728452100792715788562194946, −11.67933683653465055000978727371, −10.89751449404745670379350266001, −9.93128278941798033587352333425, −8.58985674423033362257128680458, −7.69982236300385376475402950084, −7.10030920515792657006298041478, −5.415440216554138967848830154320, −4.82341954659545789035769682235, −3.28441343474928428586314774438, −2.48070509368867572899325213004, −0.729361763714265569174132607086, 0.72710152732395946722029175300, 2.03468795195088243858968970930, 3.53899234216175068095752437585, 4.64966266249636659331201476750, 5.27354146069274675591417810269, 6.90360778629210760392188172363, 7.94678418198404842412059325642, 8.34150223179178352788705618169, 9.83002212020972549221757779464, 10.70351518157182673656033943412, 11.822316356278014853531920267785, 12.39649915766377880912342953843, 13.54680521840761722792106317308, 14.63259391933379625261318278323, 15.282297361709680356091509720855, 16.50679226372460606339714420259, 16.97779089809865510595710142056, 18.2196183856438819710855257822, 18.99529135092401591517480947049, 20.0257335900005584634729717086, 20.755544104832604904347612619, 21.41937806906940832086494847327, 22.73075088594308913483234372883, 23.57016316199371740127413884620, 24.11625517500847452493592739328

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