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Properties

Label 2-483-23.22-c2-0-21
Degree $2$
Conductor $483$
Sign $0.986 + 0.160i$
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  + 1.38·2-s − 1.73·3-s − 2.07·4-s + 1.23i·5-s − 2.40·6-s − 2.64i·7-s − 8.43·8-s + 2.99·9-s + 1.71i·10-s + 2.27i·11-s + 3.58·12-s + 15.5·13-s − 3.67i·14-s − 2.14i·15-s − 3.42·16-s − 0.389i·17-s + ⋯
L(s)  = 1  + 0.694·2-s − 0.577·3-s − 0.517·4-s + 0.247i·5-s − 0.400·6-s − 0.377i·7-s − 1.05·8-s + 0.333·9-s + 0.171i·10-s + 0.206i·11-s + 0.299·12-s + 1.19·13-s − 0.262i·14-s − 0.143i·15-s − 0.213·16-s − 0.0229i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.986 + 0.160i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.986 + 0.160i$
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: \((2,\ 483,\ (\ :1),\ 0.986 + 0.160i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.690374911\)
\(L(\frac12)\) \(\approx\) \(1.690374911\)
\(L(2)\) 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 \( 1 + (-3.70 + 22.7i)T \)
good2 \( 1 - 1.38T + 4T^{2} \)
5 \( 1 - 1.23iT - 25T^{2} \)
11 \( 1 - 2.27iT - 121T^{2} \)
13 \( 1 - 15.5T + 169T^{2} \)
17 \( 1 + 0.389iT - 289T^{2} \)
19 \( 1 - 3.84iT - 361T^{2} \)
29 \( 1 - 39.3T + 841T^{2} \)
31 \( 1 - 31.1T + 961T^{2} \)
37 \( 1 + 38.5iT - 1.36e3T^{2} \)
41 \( 1 - 19.0T + 1.68e3T^{2} \)
43 \( 1 - 79.0iT - 1.84e3T^{2} \)
47 \( 1 - 0.242T + 2.20e3T^{2} \)
53 \( 1 - 0.762iT - 2.80e3T^{2} \)
59 \( 1 - 31.8T + 3.48e3T^{2} \)
61 \( 1 + 114. iT - 3.72e3T^{2} \)
67 \( 1 - 85.0iT - 4.48e3T^{2} \)
71 \( 1 + 75.4T + 5.04e3T^{2} \)
73 \( 1 - 41.2T + 5.32e3T^{2} \)
79 \( 1 + 2.88iT - 6.24e3T^{2} \)
83 \( 1 + 105. iT - 6.88e3T^{2} \)
89 \( 1 + 71.8iT - 7.92e3T^{2} \)
97 \( 1 - 90.0iT - 9.40e3T^{2} \)
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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

−10.81750831917419745713775837763, −10.05207322427716612346793453036, −8.945996409312616403717758633937, −8.079835810155008532119863963892, −6.66720437838146242792837290484, −6.07653878418991624993158501055, −4.88100823238745784885545396179, −4.15396270688303551624108352283, −2.96558594550365448126171720379, −0.869155131160972351655898386049, 0.980814669224816011249206435929, 3.02561804615958781155363256913, 4.16558959570540534337272589878, 5.11372743376134449810201066248, 5.89214820413300717366983197571, 6.76647833983548036371239774969, 8.316563918050372468544104025294, 8.898596004391858929631673759239, 9.959088177975736636092705745322, 10.96104204588628118346202467389

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