Properties

Label 2-483-161.19-c1-0-27
Degree $2$
Conductor $483$
Sign $0.489 + 0.871i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (1.10 + 0.105i)2-s + (−0.690 + 0.723i)3-s + (−0.750 − 0.144i)4-s + (3.37 − 1.73i)5-s + (−0.840 + 0.727i)6-s + (−0.627 − 2.57i)7-s + (−2.94 − 0.865i)8-s + (−0.0475 − 0.998i)9-s + (3.91 − 1.56i)10-s + (−0.285 − 2.99i)11-s + (0.622 − 0.443i)12-s + (−2.29 − 0.329i)13-s + (−0.422 − 2.91i)14-s + (−1.06 + 3.63i)15-s + (−1.75 − 0.701i)16-s + (1.19 − 3.46i)17-s + ⋯
L(s)  = 1  + (0.782 + 0.0747i)2-s + (−0.398 + 0.417i)3-s + (−0.375 − 0.0723i)4-s + (1.50 − 0.777i)5-s + (−0.342 + 0.297i)6-s + (−0.237 − 0.971i)7-s + (−1.04 − 0.306i)8-s + (−0.0158 − 0.332i)9-s + (1.23 − 0.495i)10-s + (−0.0861 − 0.902i)11-s + (0.179 − 0.127i)12-s + (−0.636 − 0.0914i)13-s + (−0.112 − 0.777i)14-s + (−0.275 + 0.939i)15-s + (−0.437 − 0.175i)16-s + (0.290 − 0.839i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.489 + 0.871i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.489 + 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.53408 - 0.897892i\)
\(L(\frac12)\) \(\approx\) \(1.53408 - 0.897892i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.690 - 0.723i)T \)
7 \( 1 + (0.627 + 2.57i)T \)
23 \( 1 + (1.59 + 4.52i)T \)
good2 \( 1 + (-1.10 - 0.105i)T + (1.96 + 0.378i)T^{2} \)
5 \( 1 + (-3.37 + 1.73i)T + (2.90 - 4.07i)T^{2} \)
11 \( 1 + (0.285 + 2.99i)T + (-10.8 + 2.08i)T^{2} \)
13 \( 1 + (2.29 + 0.329i)T + (12.4 + 3.66i)T^{2} \)
17 \( 1 + (-1.19 + 3.46i)T + (-13.3 - 10.5i)T^{2} \)
19 \( 1 + (-2.28 - 6.58i)T + (-14.9 + 11.7i)T^{2} \)
29 \( 1 + (-6.18 - 7.13i)T + (-4.12 + 28.7i)T^{2} \)
31 \( 1 + (-2.57 + 0.625i)T + (27.5 - 14.2i)T^{2} \)
37 \( 1 + (2.24 - 0.107i)T + (36.8 - 3.51i)T^{2} \)
41 \( 1 + (-3.73 + 5.80i)T + (-17.0 - 37.2i)T^{2} \)
43 \( 1 + (-1.76 - 6.00i)T + (-36.1 + 23.2i)T^{2} \)
47 \( 1 + (7.19 + 4.15i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-8.24 - 10.4i)T + (-12.4 + 51.5i)T^{2} \)
59 \( 1 + (-0.639 - 1.59i)T + (-42.7 + 40.7i)T^{2} \)
61 \( 1 + (-5.01 + 4.78i)T + (2.90 - 60.9i)T^{2} \)
67 \( 1 + (6.05 + 4.30i)T + (21.9 + 63.3i)T^{2} \)
71 \( 1 + (-3.77 - 8.26i)T + (-46.4 + 53.6i)T^{2} \)
73 \( 1 + (-2.35 + 12.2i)T + (-67.7 - 27.1i)T^{2} \)
79 \( 1 + (7.26 - 9.24i)T + (-18.6 - 76.7i)T^{2} \)
83 \( 1 + (11.3 - 7.28i)T + (34.4 - 75.4i)T^{2} \)
89 \( 1 + (0.831 - 3.42i)T + (-79.1 - 40.7i)T^{2} \)
97 \( 1 + (-3.50 - 2.25i)T + (40.2 + 88.2i)T^{2} \)
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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

−10.54822794757851844309429659669, −9.976106339352157889279900399087, −9.308808521103431690702015066750, −8.312486588069739262757546437607, −6.70724799578779485620054519234, −5.79128451773759179100162395620, −5.19764928004462766069000856920, −4.30192242733012237282024845458, −3.03372569952036896299266527374, −0.932316923331773134339088621892, 2.13759028843115842651420022507, 2.93134631182516558161414650565, 4.70167295758669021955451168531, 5.55980895857312662854988650876, 6.20113019262986896710172664296, 7.10605555645659576612919021322, 8.556584459324168797014789487284, 9.694033101417849464754601473436, 9.959243829627698355697803076316, 11.43553645767493061579153641452

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