Properties

Label 2-483-23.22-c2-0-33
Degree $2$
Conductor $483$
Sign $0.450 + 0.892i$
Analytic cond. $13.1607$
Root an. cond. $3.62778$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.934·2-s + 1.73·3-s − 3.12·4-s − 0.661i·5-s + 1.61·6-s + 2.64i·7-s − 6.65·8-s + 2.99·9-s − 0.617i·10-s − 14.7i·11-s − 5.41·12-s + 10.6·13-s + 2.47i·14-s − 1.14i·15-s + 6.29·16-s − 15.7i·17-s + ⋯
L(s)  = 1  + 0.467·2-s + 0.577·3-s − 0.781·4-s − 0.132i·5-s + 0.269·6-s + 0.377i·7-s − 0.832·8-s + 0.333·9-s − 0.0617i·10-s − 1.33i·11-s − 0.451·12-s + 0.817·13-s + 0.176i·14-s − 0.0763i·15-s + 0.393·16-s − 0.927i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.450 + 0.892i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.450 + 0.892i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (22, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ 0.450 + 0.892i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.025685729\)
\(L(\frac12)\) \(\approx\) \(2.025685729\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 - 1.73T \)
7 \( 1 - 2.64iT \)
23 \( 1 + (-20.5 + 10.3i)T \)
good2 \( 1 - 0.934T + 4T^{2} \)
5 \( 1 + 0.661iT - 25T^{2} \)
11 \( 1 + 14.7iT - 121T^{2} \)
13 \( 1 - 10.6T + 169T^{2} \)
17 \( 1 + 15.7iT - 289T^{2} \)
19 \( 1 + 14.9iT - 361T^{2} \)
29 \( 1 + 22.7T + 841T^{2} \)
31 \( 1 + 1.01T + 961T^{2} \)
37 \( 1 + 36.9iT - 1.36e3T^{2} \)
41 \( 1 + 32.4T + 1.68e3T^{2} \)
43 \( 1 + 0.910iT - 1.84e3T^{2} \)
47 \( 1 - 21.9T + 2.20e3T^{2} \)
53 \( 1 + 20.2iT - 2.80e3T^{2} \)
59 \( 1 + 47.8T + 3.48e3T^{2} \)
61 \( 1 + 40.5iT - 3.72e3T^{2} \)
67 \( 1 - 70.4iT - 4.48e3T^{2} \)
71 \( 1 - 29.6T + 5.04e3T^{2} \)
73 \( 1 - 13.1T + 5.32e3T^{2} \)
79 \( 1 - 84.4iT - 6.24e3T^{2} \)
83 \( 1 + 61.4iT - 6.88e3T^{2} \)
89 \( 1 - 93.0iT - 7.92e3T^{2} \)
97 \( 1 + 34.1iT - 9.40e3T^{2} \)
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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

−10.72511019505678835029966635646, −9.322554948369854596389406800534, −8.906872146906250694268468209332, −8.209908363685896555587602341686, −6.85236931124116265698107753087, −5.70366324591853593946474636389, −4.86372750127420255588497633042, −3.64436220365908600516749195014, −2.81485253329867431353163095966, −0.73843247786484363496580704771, 1.51929679973589587454126514419, 3.23417437336664439617729833265, 4.08934565106977240699758636384, 5.00409121481149978135193775250, 6.23731055704034668380938478793, 7.33428460789247681646857511613, 8.309131684705469536730547666077, 9.127875485871410967193977272349, 9.968491552555316416173280010906, 10.77186117257386907020885484667

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