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Properties

Label 1-1536-1536.515-r0-0-0
Degree $1$
Conductor $1536$
Sign $0.134 - 0.990i$
Analytic cond. $7.13315$
Root an. cond. $7.13315$
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  + (0.146 − 0.989i)5-s + (−0.0980 − 0.995i)7-s + (0.0490 + 0.998i)11-s + (−0.595 + 0.803i)13-s + (0.555 + 0.831i)17-s + (0.242 − 0.970i)19-s + (0.471 − 0.881i)23-s + (−0.956 − 0.290i)25-s + (0.671 + 0.740i)29-s + (−0.382 − 0.923i)31-s + (−0.998 − 0.0490i)35-s + (0.514 − 0.857i)37-s + (0.956 − 0.290i)41-s + (0.427 + 0.903i)43-s + (0.980 + 0.195i)47-s + ⋯
L(s)  = 1  + (0.146 − 0.989i)5-s + (−0.0980 − 0.995i)7-s + (0.0490 + 0.998i)11-s + (−0.595 + 0.803i)13-s + (0.555 + 0.831i)17-s + (0.242 − 0.970i)19-s + (0.471 − 0.881i)23-s + (−0.956 − 0.290i)25-s + (0.671 + 0.740i)29-s + (−0.382 − 0.923i)31-s + (−0.998 − 0.0490i)35-s + (0.514 − 0.857i)37-s + (0.956 − 0.290i)41-s + (0.427 + 0.903i)43-s + (0.980 + 0.195i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.134 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.134 - 0.990i$
Analytic conductor: \(7.13315\)
Root analytic conductor: \(7.13315\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 1536,\ (0:\ ),\ 0.134 - 0.990i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.122475103 - 0.9803300280i\)
\(L(\frac12)\) \(\approx\) \(1.122475103 - 0.9803300280i\)
\(L(1)\) \(\approx\) \(1.044660975 - 0.3118241077i\)
\(L(1)\) \(\approx\) \(1.044660975 - 0.3118241077i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
good5 \( 1 + (0.146 - 0.989i)T \)
7 \( 1 + (-0.0980 - 0.995i)T \)
11 \( 1 + (0.0490 + 0.998i)T \)
13 \( 1 + (-0.595 + 0.803i)T \)
17 \( 1 + (0.555 + 0.831i)T \)
19 \( 1 + (0.242 - 0.970i)T \)
23 \( 1 + (0.471 - 0.881i)T \)
29 \( 1 + (0.671 + 0.740i)T \)
31 \( 1 + (-0.382 - 0.923i)T \)
37 \( 1 + (0.514 - 0.857i)T \)
41 \( 1 + (0.956 - 0.290i)T \)
43 \( 1 + (0.427 + 0.903i)T \)
47 \( 1 + (0.980 + 0.195i)T \)
53 \( 1 + (0.671 - 0.740i)T \)
59 \( 1 + (0.595 + 0.803i)T \)
61 \( 1 + (-0.336 - 0.941i)T \)
67 \( 1 + (0.941 - 0.336i)T \)
71 \( 1 + (-0.773 - 0.634i)T \)
73 \( 1 + (0.0980 - 0.995i)T \)
79 \( 1 + (-0.195 - 0.980i)T \)
83 \( 1 + (0.514 + 0.857i)T \)
89 \( 1 + (-0.471 - 0.881i)T \)
97 \( 1 + (-0.923 + 0.382i)T \)
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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

−20.92200783743534375664803752135, −19.85306968453749317709161398724, −19.00349691609551508543370049822, −18.65005746542963386905674434106, −17.88527474877042779558513765843, −17.10845180105597054462607732575, −16.06178815852302632010036054572, −15.5057204317867803264615627165, −14.6714593909190304585130636317, −14.08735151876221117630157759468, −13.27674072231719056291293435697, −12.20167565910848251054599315043, −11.69103537845901313488096989155, −10.8090344770141220848119649632, −10.009273865128575863107781794372, −9.32944188655706137139178630853, −8.30506914315517680342123736323, −7.581274240970216195495141482810, −6.6900638418379645773724484234, −5.61840395894978972834532273677, −5.46264997782730730071998611046, −3.86571877965703528840097276837, −2.923978358068308183117631031363, −2.56516904918075288376550465831, −1.10714705480865290785223151553, 0.63515533634841437862078545283, 1.61374311510126261858116328504, 2.59403118919182150270631699327, 4.049133352213356732677880780808, 4.44816918322049410757011582589, 5.287684161329452200021870038311, 6.42659268375611904227591273082, 7.23930291156139334823652808568, 7.90388338604644564722827804002, 9.02835431585597509066864683974, 9.54029720174307614634905527742, 10.384655561716606114590174051677, 11.217147527768292738676087835275, 12.31512294366757402605000680082, 12.71117542244390691814876796713, 13.52884967366208645526391687467, 14.38016943274349241657921860925, 15.03300360272512150292032256791, 16.20203641643194997152737926138, 16.62382162644975769127914641192, 17.3783407287764918238549776960, 17.88001825920749401203088484885, 19.16747993235778853739226063109, 19.74888684231817429432021875507, 20.35087342273890009041073085886

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