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Properties

Label 1-3864-3864.443-r0-0-0
Degree $1$
Conductor $3864$
Sign $0.751 - 0.659i$
Analytic cond. $17.9443$
Root an. cond. $17.9443$
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  + (−0.995 + 0.0950i)5-s + (0.327 + 0.945i)11-s + (0.959 − 0.281i)13-s + (−0.928 − 0.371i)17-s + (0.928 − 0.371i)19-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.580 + 0.814i)37-s + (−0.415 + 0.909i)41-s + (0.841 + 0.540i)43-s + (−0.5 − 0.866i)47-s + (0.723 − 0.690i)53-s + (−0.415 − 0.909i)55-s + (−0.235 + 0.971i)59-s + ⋯
L(s)  = 1  + (−0.995 + 0.0950i)5-s + (0.327 + 0.945i)11-s + (0.959 − 0.281i)13-s + (−0.928 − 0.371i)17-s + (0.928 − 0.371i)19-s + (0.981 − 0.189i)25-s + (−0.142 − 0.989i)29-s + (−0.0475 − 0.998i)31-s + (−0.580 + 0.814i)37-s + (−0.415 + 0.909i)41-s + (0.841 + 0.540i)43-s + (−0.5 − 0.866i)47-s + (0.723 − 0.690i)53-s + (−0.415 − 0.909i)55-s + (−0.235 + 0.971i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3864 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3864\)    =    \(2^{3} \cdot 3 \cdot 7 \cdot 23\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(17.9443\)
Root analytic conductor: \(17.9443\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3864} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3864,\ (0:\ ),\ 0.751 - 0.659i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.194045428 - 0.4499626059i\)
\(L(\frac12)\) \(\approx\) \(1.194045428 - 0.4499626059i\)
\(L(1)\) \(\approx\) \(0.9316211225 - 0.03493531605i\)
\(L(1)\) \(\approx\) \(0.9316211225 - 0.03493531605i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
7 \( 1 \)
23 \( 1 \)
good5 \( 1 + (-0.995 + 0.0950i)T \)
11 \( 1 + (0.327 + 0.945i)T \)
13 \( 1 + (0.959 - 0.281i)T \)
17 \( 1 + (-0.928 - 0.371i)T \)
19 \( 1 + (0.928 - 0.371i)T \)
29 \( 1 + (-0.142 - 0.989i)T \)
31 \( 1 + (-0.0475 - 0.998i)T \)
37 \( 1 + (-0.580 + 0.814i)T \)
41 \( 1 + (-0.415 + 0.909i)T \)
43 \( 1 + (0.841 + 0.540i)T \)
47 \( 1 + (-0.5 - 0.866i)T \)
53 \( 1 + (0.723 - 0.690i)T \)
59 \( 1 + (-0.235 + 0.971i)T \)
61 \( 1 + (0.888 + 0.458i)T \)
67 \( 1 + (0.981 - 0.189i)T \)
71 \( 1 + (-0.654 - 0.755i)T \)
73 \( 1 + (-0.786 - 0.618i)T \)
79 \( 1 + (-0.723 - 0.690i)T \)
83 \( 1 + (-0.415 - 0.909i)T \)
89 \( 1 + (-0.0475 + 0.998i)T \)
97 \( 1 + (0.415 - 0.909i)T \)
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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

−18.85154697259911223306713749091, −17.99478915650803492873032289105, −17.27054309385480513889105370376, −16.31457230991389945092620501174, −15.95654892040334162423363136697, −15.47659255611494518184551068793, −14.332405470786579743732372412651, −14.04361209637492681033302766685, −13.0683190138397785543997587304, −12.41299436821430799324778271701, −11.66328050991342347052691500743, −11.01433783245344753572201853306, −10.64440390924912928063192280059, −9.41569195029361422625024570369, −8.586403463011926159180009122486, −8.4465071401881537355930097731, −7.286289752779994188280020598917, −6.795288227938491343162101388454, −5.84054505425475588788860014836, −5.1444815422926894353169854060, −4.057099749306182680956650506225, −3.66676861137926168689315284122, −2.89256905112523516962330841864, −1.62627698610770100735935917249, −0.82913250023151866523444612144, 0.49301666153626838754575692307, 1.53329703243339350372992358493, 2.5761676904459978292424204280, 3.39092867573510049023863429320, 4.20371168156407639940742711613, 4.70204555104107171067391863237, 5.717468477000566833194381638382, 6.644650372385986160573403930664, 7.20776935023743712456370803236, 7.95113528852594439377252474903, 8.62362167719511591204567796819, 9.41663420077917918030011663401, 10.13978614824496022730744635163, 11.04681290524308399112086892393, 11.64613764295892797703275390134, 12.030607029349444299655276127017, 13.211427153787574169463324532237, 13.40610224813951572030532083386, 14.58153697474718138216545325174, 15.159409609037389312155878767, 15.694466502896195448751099368470, 16.26734987641305833978775051140, 17.13450030314880632491909257434, 17.94301983456282778415163532202, 18.36699326750509975502394872471

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