Properties

Label 2-483-161.160-c1-0-10
Degree $2$
Conductor $483$
Sign $0.694 - 0.719i$
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  + 0.254·2-s i·3-s − 1.93·4-s − 0.458·5-s − 0.254i·6-s + (1.07 + 2.41i)7-s − 8-s − 9-s − 0.116·10-s + 1.34i·11-s + 1.93i·12-s + 4.42i·13-s + (0.272 + 0.614i)14-s + 0.458i·15-s + 3.61·16-s + 7.09·17-s + ⋯
L(s)  = 1  + 0.179·2-s − 0.577i·3-s − 0.967·4-s − 0.204·5-s − 0.103i·6-s + (0.405 + 0.914i)7-s − 0.353·8-s − 0.333·9-s − 0.0368·10-s + 0.405i·11-s + 0.558i·12-s + 1.22i·13-s + (0.0728 + 0.164i)14-s + 0.118i·15-s + 0.904·16-s + 1.72·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.694 - 0.719i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.694 - 0.719i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.694 - 0.719i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04673 + 0.444512i\)
\(L(\frac12)\) \(\approx\) \(1.04673 + 0.444512i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-1.07 - 2.41i)T \)
23 \( 1 + (4.44 - 1.80i)T \)
good2 \( 1 - 0.254T + 2T^{2} \)
5 \( 1 + 0.458T + 5T^{2} \)
11 \( 1 - 1.34iT - 11T^{2} \)
13 \( 1 - 4.42iT - 13T^{2} \)
17 \( 1 - 7.09T + 17T^{2} \)
19 \( 1 - 4.40T + 19T^{2} \)
29 \( 1 - 4.18T + 29T^{2} \)
31 \( 1 + 4.31iT - 31T^{2} \)
37 \( 1 - 8.56iT - 37T^{2} \)
41 \( 1 - 3.81iT - 41T^{2} \)
43 \( 1 - 7.18iT - 43T^{2} \)
47 \( 1 - 7.23iT - 47T^{2} \)
53 \( 1 + 8.35iT - 53T^{2} \)
59 \( 1 + 8.10iT - 59T^{2} \)
61 \( 1 + 10.2T + 61T^{2} \)
67 \( 1 - 5.84iT - 67T^{2} \)
71 \( 1 - 8.42T + 71T^{2} \)
73 \( 1 + 7.04iT - 73T^{2} \)
79 \( 1 - 6.86iT - 79T^{2} \)
83 \( 1 + 4.95T + 83T^{2} \)
89 \( 1 - 16.0T + 89T^{2} \)
97 \( 1 + 3.86T + 97T^{2} \)
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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

−11.58275220935982815930709406185, −9.793512515238799584277929428817, −9.478485507046573362245390835073, −8.175314363533576140029181655507, −7.78817623485236255135822487363, −6.30027767146910708281340740530, −5.42615975649888714392222794534, −4.45224955099655560302130781008, −3.16135255178487280139185860636, −1.53335535957880372390536042589, 0.75318145690819381383556658364, 3.31076798953150668395354237144, 3.95564213444854705389079278858, 5.16541029374063012178142135263, 5.76922266727565427745632788641, 7.58093660426084873950626414041, 8.076028601693881119378159269884, 9.146891819805436966522595405368, 10.28344925911979451542774731037, 10.40041160413968983857020854852

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