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Properties

Label 2-483-69.68-c1-0-10
Degree $2$
Conductor $483$
Sign $0.789 - 0.613i$
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  + 1.06i·2-s + (−1.10 − 1.33i)3-s + 0.872·4-s − 4.21·5-s + (1.41 − 1.16i)6-s − i·7-s + 3.05i·8-s + (−0.576 + 2.94i)9-s − 4.47i·10-s + 2.41·11-s + (−0.960 − 1.16i)12-s + 4.65·13-s + 1.06·14-s + (4.63 + 5.63i)15-s − 1.49·16-s + 3.38·17-s + ⋯
L(s)  = 1  + 0.750i·2-s + (−0.635 − 0.772i)3-s + 0.436·4-s − 1.88·5-s + (0.579 − 0.477i)6-s − 0.377i·7-s + 1.07i·8-s + (−0.192 + 0.981i)9-s − 1.41i·10-s + 0.727·11-s + (−0.277 − 0.336i)12-s + 1.29·13-s + 0.283·14-s + (1.19 + 1.45i)15-s − 0.373·16-s + 0.821·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.789 - 0.613i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.789 - 0.613i$
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: \((2,\ 483,\ (\ :1/2),\ 0.789 - 0.613i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.993621 + 0.340728i\)
\(L(\frac12)\) \(\approx\) \(0.993621 + 0.340728i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.10 + 1.33i)T \)
7 \( 1 + iT \)
23 \( 1 + (-4.79 - 0.134i)T \)
good2 \( 1 - 1.06iT - 2T^{2} \)
5 \( 1 + 4.21T + 5T^{2} \)
11 \( 1 - 2.41T + 11T^{2} \)
13 \( 1 - 4.65T + 13T^{2} \)
17 \( 1 - 3.38T + 17T^{2} \)
19 \( 1 - 0.938iT - 19T^{2} \)
29 \( 1 + 2.21iT - 29T^{2} \)
31 \( 1 + 0.993T + 31T^{2} \)
37 \( 1 + 0.131iT - 37T^{2} \)
41 \( 1 + 7.16iT - 41T^{2} \)
43 \( 1 - 4.89iT - 43T^{2} \)
47 \( 1 - 7.21iT - 47T^{2} \)
53 \( 1 + 2.35T + 53T^{2} \)
59 \( 1 - 12.6iT - 59T^{2} \)
61 \( 1 - 4.95iT - 61T^{2} \)
67 \( 1 - 4.58iT - 67T^{2} \)
71 \( 1 + 16.5iT - 71T^{2} \)
73 \( 1 - 3.32T + 73T^{2} \)
79 \( 1 - 8.91iT - 79T^{2} \)
83 \( 1 + 3.38T + 83T^{2} \)
89 \( 1 - 15.6T + 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−11.17320226638595132221839437508, −10.74340154479458538050858699058, −8.762855414535410870487497749735, −7.973895193009695621683510171180, −7.38452635766581933019748109216, −6.68751171173333782681524151031, −5.74543479505129999574323263396, −4.43928464101079898694346992510, −3.26009283606937401689697777346, −1.12314312461610211822623832378, 0.935688257951240249349991064243, 3.36257206340952889461981611226, 3.67722916660987980979317791484, 4.86721509835898990309404521607, 6.29737324463590096775141658516, 7.14028675688545096059460677347, 8.353559986070457907110006009338, 9.216162463634592285132636664967, 10.39146605610922877927876506779, 11.24423068931382046128536549139

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