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Properties

Label 2-483-23.22-c2-0-13
Degree $2$
Conductor $483$
Sign $-0.411 - 0.911i$
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  + 2.83·2-s − 1.73·3-s + 4.06·4-s + 7.71i·5-s − 4.91·6-s − 2.64i·7-s + 0.172·8-s + 2.99·9-s + 21.9i·10-s + 3.20i·11-s − 7.03·12-s + 0.727·13-s − 7.51i·14-s − 13.3i·15-s − 15.7·16-s + 15.2i·17-s + ⋯
L(s)  = 1  + 1.41·2-s − 0.577·3-s + 1.01·4-s + 1.54i·5-s − 0.819·6-s − 0.377i·7-s + 0.0215·8-s + 0.333·9-s + 2.19i·10-s + 0.291i·11-s − 0.586·12-s + 0.0559·13-s − 0.536i·14-s − 0.890i·15-s − 0.984·16-s + 0.895i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.411 - 0.911i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.411 - 0.911i$
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: \((2,\ 483,\ (\ :1),\ -0.411 - 0.911i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.325769914\)
\(L(\frac12)\) \(\approx\) \(2.325769914\)
\(L(2)\) 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 \( 1 + (20.9 - 9.45i)T \)
good2 \( 1 - 2.83T + 4T^{2} \)
5 \( 1 - 7.71iT - 25T^{2} \)
11 \( 1 - 3.20iT - 121T^{2} \)
13 \( 1 - 0.727T + 169T^{2} \)
17 \( 1 - 15.2iT - 289T^{2} \)
19 \( 1 - 24.4iT - 361T^{2} \)
29 \( 1 - 11.3T + 841T^{2} \)
31 \( 1 + 29.1T + 961T^{2} \)
37 \( 1 - 44.8iT - 1.36e3T^{2} \)
41 \( 1 - 73.8T + 1.68e3T^{2} \)
43 \( 1 + 59.9iT - 1.84e3T^{2} \)
47 \( 1 - 68.3T + 2.20e3T^{2} \)
53 \( 1 + 13.5iT - 2.80e3T^{2} \)
59 \( 1 + 37.9T + 3.48e3T^{2} \)
61 \( 1 + 31.5iT - 3.72e3T^{2} \)
67 \( 1 + 36.0iT - 4.48e3T^{2} \)
71 \( 1 - 102.T + 5.04e3T^{2} \)
73 \( 1 - 124.T + 5.32e3T^{2} \)
79 \( 1 - 76.5iT - 6.24e3T^{2} \)
83 \( 1 - 3.13iT - 6.88e3T^{2} \)
89 \( 1 - 48.0iT - 7.92e3T^{2} \)
97 \( 1 - 155. iT - 9.40e3T^{2} \)
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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

−11.08713683089932450798366579908, −10.58727328174787228633211906787, −9.663190060920291931961210549469, −7.938192896110057074264052608437, −6.95284133686600293634946188701, −6.22152991640645280870553927381, −5.53022722468192424714630781237, −4.10772283666753263385687028080, −3.51675533318817838842029752177, −2.15091100069938702300879782224, 0.61562796509819309962697362452, 2.47263413847798611845411599301, 4.05697150881190835737640750306, 4.80566993577150262369880298548, 5.48099365502317707643828373629, 6.26093571588815252274675489819, 7.55617388234298656350421897226, 8.906757403933565902017865228942, 9.375712213733291911349618004862, 10.97171843409684257520783165686

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