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Properties

Label 2-483-3.2-c2-0-10
Degree $2$
Conductor $483$
Sign $-0.966 - 0.258i$
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  + 1.72i·2-s + (0.774 − 2.89i)3-s + 1.01·4-s + 6.73i·5-s + (5.00 + 1.33i)6-s − 2.64·7-s + 8.66i·8-s + (−7.79 − 4.49i)9-s − 11.6·10-s + 18.2i·11-s + (0.784 − 2.93i)12-s − 16.7·13-s − 4.57i·14-s + (19.5 + 5.21i)15-s − 10.9·16-s − 27.0i·17-s + ⋯
L(s)  = 1  + 0.864i·2-s + (0.258 − 0.966i)3-s + 0.253·4-s + 1.34i·5-s + (0.834 + 0.223i)6-s − 0.377·7-s + 1.08i·8-s + (−0.866 − 0.498i)9-s − 1.16·10-s + 1.66i·11-s + (0.0654 − 0.244i)12-s − 1.28·13-s − 0.326i·14-s + (1.30 + 0.347i)15-s − 0.682·16-s − 1.59i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.966 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.966 - 0.258i$
Analytic conductor: \(13.1607\)
Root analytic conductor: \(3.62778\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1),\ -0.966 - 0.258i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.219264694\)
\(L(\frac12)\) \(\approx\) \(1.219264694\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.774 + 2.89i)T \)
7 \( 1 + 2.64T \)
23 \( 1 - 4.79iT \)
good2 \( 1 - 1.72iT - 4T^{2} \)
5 \( 1 - 6.73iT - 25T^{2} \)
11 \( 1 - 18.2iT - 121T^{2} \)
13 \( 1 + 16.7T + 169T^{2} \)
17 \( 1 + 27.0iT - 289T^{2} \)
19 \( 1 + 24.6T + 361T^{2} \)
29 \( 1 + 18.0iT - 841T^{2} \)
31 \( 1 - 15.6T + 961T^{2} \)
37 \( 1 - 5.09T + 1.36e3T^{2} \)
41 \( 1 - 54.0iT - 1.68e3T^{2} \)
43 \( 1 - 24.0T + 1.84e3T^{2} \)
47 \( 1 - 20.5iT - 2.20e3T^{2} \)
53 \( 1 - 39.1iT - 2.80e3T^{2} \)
59 \( 1 - 43.3iT - 3.48e3T^{2} \)
61 \( 1 + 29.0T + 3.72e3T^{2} \)
67 \( 1 - 126.T + 4.48e3T^{2} \)
71 \( 1 - 120. iT - 5.04e3T^{2} \)
73 \( 1 + 14.2T + 5.32e3T^{2} \)
79 \( 1 - 21.6T + 6.24e3T^{2} \)
83 \( 1 + 27.7iT - 6.88e3T^{2} \)
89 \( 1 + 166. iT - 7.92e3T^{2} \)
97 \( 1 - 134.T + 9.40e3T^{2} \)
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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

−11.35517601171178045886402763905, −10.19765334099792762012658624428, −9.361129480311161475806802579329, −7.925224902357656403011775999588, −7.19370626388649775123241943629, −6.93539656794776124587442158654, −6.08899532503601474144256449981, −4.71582266553436223590709520523, −2.71723940119940725561270701173, −2.30158683567764772500815150048, 0.42684094683203585715901987806, 2.14486234134637937757728702129, 3.42097414324662736384111714892, 4.25608405099148364817122775244, 5.40253570694857691532258802919, 6.41405929654684541091668540552, 8.138392784772626040072378584832, 8.752899242141675705653944645496, 9.579506477881354000386328210292, 10.52073743390799301425505438576

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