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Properties

Label 1-3840-3840.2243-r1-0-0
Degree $1$
Conductor $3840$
Sign $0.714 - 0.699i$
Analytic cond. $412.665$
Root an. cond. $412.665$
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  + (0.195 + 0.980i)7-s + (0.0980 + 0.995i)11-s + (0.290 + 0.956i)13-s + (−0.923 − 0.382i)17-s + (0.471 − 0.881i)19-s + (−0.831 + 0.555i)23-s + (−0.995 − 0.0980i)29-s + (0.707 − 0.707i)31-s + (0.471 + 0.881i)37-s + (0.831 − 0.555i)41-s + (0.634 − 0.773i)43-s + (0.382 − 0.923i)47-s + (−0.923 + 0.382i)49-s + (0.0980 + 0.995i)53-s + (−0.956 − 0.290i)59-s + ⋯
L(s)  = 1  + (0.195 + 0.980i)7-s + (0.0980 + 0.995i)11-s + (0.290 + 0.956i)13-s + (−0.923 − 0.382i)17-s + (0.471 − 0.881i)19-s + (−0.831 + 0.555i)23-s + (−0.995 − 0.0980i)29-s + (0.707 − 0.707i)31-s + (0.471 + 0.881i)37-s + (0.831 − 0.555i)41-s + (0.634 − 0.773i)43-s + (0.382 − 0.923i)47-s + (−0.923 + 0.382i)49-s + (0.0980 + 0.995i)53-s + (−0.956 − 0.290i)59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3840 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.714 - 0.699i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(3840\)    =    \(2^{8} \cdot 3 \cdot 5\)
Sign: $0.714 - 0.699i$
Analytic conductor: \(412.665\)
Root analytic conductor: \(412.665\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3840} (2243, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 3840,\ (1:\ ),\ 0.714 - 0.699i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.271895361 - 0.5187107016i\)
\(L(\frac12)\) \(\approx\) \(1.271895361 - 0.5187107016i\)
\(L(1)\) \(\approx\) \(0.9893799933 + 0.1490215722i\)
\(L(1)\) \(\approx\) \(0.9893799933 + 0.1490215722i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 \)
good7 \( 1 + (0.195 + 0.980i)T \)
11 \( 1 + (0.0980 + 0.995i)T \)
13 \( 1 + (0.290 + 0.956i)T \)
17 \( 1 + (-0.923 - 0.382i)T \)
19 \( 1 + (0.471 - 0.881i)T \)
23 \( 1 + (-0.831 + 0.555i)T \)
29 \( 1 + (-0.995 - 0.0980i)T \)
31 \( 1 + (0.707 - 0.707i)T \)
37 \( 1 + (0.471 + 0.881i)T \)
41 \( 1 + (0.831 - 0.555i)T \)
43 \( 1 + (0.634 - 0.773i)T \)
47 \( 1 + (0.382 - 0.923i)T \)
53 \( 1 + (0.0980 + 0.995i)T \)
59 \( 1 + (-0.956 - 0.290i)T \)
61 \( 1 + (-0.634 - 0.773i)T \)
67 \( 1 + (-0.773 + 0.634i)T \)
71 \( 1 + (-0.195 - 0.980i)T \)
73 \( 1 + (-0.195 + 0.980i)T \)
79 \( 1 + (-0.923 + 0.382i)T \)
83 \( 1 + (0.471 - 0.881i)T \)
89 \( 1 + (0.555 - 0.831i)T \)
97 \( 1 + (-0.707 - 0.707i)T \)
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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

−18.27957738997292948221265022950, −17.86818038169101927195404762588, −17.13177112759936865689108580388, −16.28274796852666487483486069527, −16.03231264245099423399350282412, −14.9581593540777304640036509281, −14.31299351799598997121102485836, −13.6999185051862640507732334458, −13.06153810607860756307262656434, −12.38621739815320325620784015417, −11.39049205427101962187331773025, −10.808767194669896438569319925568, −10.352913396919567442894889891873, −9.44364690397321396971667398472, −8.594548379022841642791091275486, −7.89453840742983137062276413067, −7.42126961891301141940338501692, −6.19559316919897064862923334908, −5.97126651634253735742977749757, −4.83526477942385742377100169267, −4.043465789657140051072083038815, −3.44126195209217032537391493896, −2.54391881908876438845445779, −1.397576125592089724868546003449, −0.72760621084266504824398858521, 0.25336816917394219714406887545, 1.62399030622156927015638771175, 2.17396634702051736333953395162, 2.94527568691469080996843752711, 4.18299604418291573260249518338, 4.587940415738290291811198093286, 5.56153709731049734038130277277, 6.23192822394146887523314023565, 7.082040561625879795315644875388, 7.67789111138127835179787138583, 8.71272185771304872999449411285, 9.2561704627056784705944719694, 9.727294561112241756179875141264, 10.81613612911704602155501456436, 11.61741246957559337805684841003, 11.901191643194046585651578289837, 12.79190661547167764080265600031, 13.57263082788861601295935290227, 14.132424352439142277049669298607, 15.16144980868929111541950815958, 15.437735883577308163060826240104, 16.11022711488252714983239399570, 17.109528811317810560793942864, 17.62901526103524710035702388061, 18.38769940547945730005335960937

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