↑

Properties

Label 2-483-21.20-c1-0-10
Degree $2$
Conductor $483$
Sign $-0.906 + 0.422i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 2.16i·2-s + (−1.71 − 0.256i)3-s − 2.67·4-s + 0.517·5-s + (0.553 − 3.70i)6-s + (1.45 + 2.20i)7-s − 1.44i·8-s + (2.86 + 0.878i)9-s + 1.11i·10-s + 4.77i·11-s + (4.57 + 0.684i)12-s − 1.14i·13-s + (−4.76 + 3.15i)14-s + (−0.886 − 0.132i)15-s − 2.20·16-s − 1.79·17-s + ⋯
L(s)  = 1  + 1.52i·2-s + (−0.988 − 0.147i)3-s − 1.33·4-s + 0.231·5-s + (0.226 − 1.51i)6-s + (0.551 + 0.833i)7-s − 0.512i·8-s + (0.956 + 0.292i)9-s + 0.353i·10-s + 1.44i·11-s + (1.32 + 0.197i)12-s − 0.317i·13-s + (−1.27 + 0.843i)14-s + (−0.228 − 0.0342i)15-s − 0.552·16-s − 0.435·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.906 + 0.422i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.906 + 0.422i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.906 + 0.422i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.184663 - 0.833687i\)
\(L(\frac12)\) \(\approx\) \(0.184663 - 0.833687i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (1.71 + 0.256i)T \)
7 \( 1 + (-1.45 - 2.20i)T \)
23 \( 1 - iT \)
good2 \( 1 - 2.16iT - 2T^{2} \)
5 \( 1 - 0.517T + 5T^{2} \)
11 \( 1 - 4.77iT - 11T^{2} \)
13 \( 1 + 1.14iT - 13T^{2} \)
17 \( 1 + 1.79T + 17T^{2} \)
19 \( 1 + 3.65iT - 19T^{2} \)
29 \( 1 + 2.33iT - 29T^{2} \)
31 \( 1 - 10.1iT - 31T^{2} \)
37 \( 1 + 2.59T + 37T^{2} \)
41 \( 1 + 9.39T + 41T^{2} \)
43 \( 1 - 10.0T + 43T^{2} \)
47 \( 1 - 2.16T + 47T^{2} \)
53 \( 1 + 0.121iT - 53T^{2} \)
59 \( 1 - 1.52T + 59T^{2} \)
61 \( 1 - 9.95iT - 61T^{2} \)
67 \( 1 - 4.00T + 67T^{2} \)
71 \( 1 + 8.01iT - 71T^{2} \)
73 \( 1 - 3.71iT - 73T^{2} \)
79 \( 1 + 14.9T + 79T^{2} \)
83 \( 1 - 15.5T + 83T^{2} \)
89 \( 1 - 9.95T + 89T^{2} \)
97 \( 1 + 3.44iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−11.62465907893858668763074932193, −10.54290235567565735572523023965, −9.498554537526329248235914722767, −8.587420020364758353992919054212, −7.49612200135948441302796633929, −6.89997845958869635246582826845, −5.94854054404298213218593016783, −5.14692695918564796628750464804, −4.52895126264754952175323988265, −2.03637368776800789862358472608, 0.59304964506342973643817174255, 1.86910519555662937586941423927, 3.59989069130470957186472802437, 4.31645094267353798000812684368, 5.54928797064391188480625484222, 6.57419053821823494634944211957, 7.88092312230054206453171639754, 9.111953632206117186807350353892, 10.06872721414518951776380042209, 10.67106182335704472554202289174

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