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Properties

Label 2-483-69.68-c1-0-30
Degree $2$
Conductor $483$
Sign $0.387 + 0.921i$
Analytic cond. $3.85677$
Root an. cond. $1.96386$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.377i·2-s + (−1.03 − 1.38i)3-s + 1.85·4-s + 2.37·5-s + (−0.523 + 0.392i)6-s − i·7-s − 1.45i·8-s + (−0.840 + 2.87i)9-s − 0.895i·10-s + 6.18·11-s + (−1.93 − 2.57i)12-s − 3.83·13-s − 0.377·14-s + (−2.46 − 3.28i)15-s + 3.16·16-s + 0.877·17-s + ⋯
L(s)  = 1  − 0.267i·2-s + (−0.599 − 0.800i)3-s + 0.928·4-s + 1.06·5-s + (−0.213 + 0.160i)6-s − 0.377i·7-s − 0.515i·8-s + (−0.280 + 0.959i)9-s − 0.283i·10-s + 1.86·11-s + (−0.557 − 0.742i)12-s − 1.06·13-s − 0.100·14-s + (−0.636 − 0.848i)15-s + 0.791·16-s + 0.212·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.387 + 0.921i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.387 + 0.921i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (344, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.387 + 0.921i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.48032 - 0.983439i\)
\(L(\frac12)\) \(\approx\) \(1.48032 - 0.983439i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.03 + 1.38i)T \)
7 \( 1 + iT \)
23 \( 1 + (1.16 - 4.65i)T \)
good2 \( 1 + 0.377iT - 2T^{2} \)
5 \( 1 - 2.37T + 5T^{2} \)
11 \( 1 - 6.18T + 11T^{2} \)
13 \( 1 + 3.83T + 13T^{2} \)
17 \( 1 - 0.877T + 17T^{2} \)
19 \( 1 - 3.29iT - 19T^{2} \)
29 \( 1 + 5.12iT - 29T^{2} \)
31 \( 1 + 8.93T + 31T^{2} \)
37 \( 1 + 10.9iT - 37T^{2} \)
41 \( 1 - 1.00iT - 41T^{2} \)
43 \( 1 - 2.83iT - 43T^{2} \)
47 \( 1 - 1.64iT - 47T^{2} \)
53 \( 1 + 4.73T + 53T^{2} \)
59 \( 1 + 10.7iT - 59T^{2} \)
61 \( 1 - 6.29iT - 61T^{2} \)
67 \( 1 - 2.23iT - 67T^{2} \)
71 \( 1 - 8.32iT - 71T^{2} \)
73 \( 1 + 9.99T + 73T^{2} \)
79 \( 1 - 13.6iT - 79T^{2} \)
83 \( 1 - 7.43T + 83T^{2} \)
89 \( 1 + 3.93T + 89T^{2} \)
97 \( 1 + 11.1iT - 97T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.02075984150405302035775455493, −9.969108770833249496135995175090, −9.357351942837371964757391932560, −7.72296437080779177341331727309, −7.03754449836347954389798954856, −6.18756690858637431584771559759, −5.55098703501156032804501683700, −3.83819417059256529281350961784, −2.18083772681073560065012617356, −1.40404223328531844167709526115, 1.75179281478078504141132368640, 3.16724118041299511939607925432, 4.66827153599803119335603821182, 5.67393683245469991343924698172, 6.42183725600406711508892611124, 7.08487974602247674511081066478, 8.771281249911480737223528379575, 9.463181167988548191557922943296, 10.23260262671187727456100445408, 11.12910144237658542947432791965

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