Properties

Label 2-483-161.61-c1-0-14
Degree $2$
Conductor $483$
Sign $-0.413 + 0.910i$
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  + (−1.77 − 1.39i)2-s + (−0.814 − 0.580i)3-s + (0.725 + 2.98i)4-s + (3.88 − 0.748i)5-s + (0.634 + 2.16i)6-s + (−2.37 + 1.16i)7-s + (1.00 − 2.20i)8-s + (0.327 + 0.945i)9-s + (−7.92 − 4.08i)10-s + (−0.933 − 1.18i)11-s + (1.14 − 2.85i)12-s + (−0.906 − 1.41i)13-s + (5.83 + 1.24i)14-s + (−3.59 − 1.64i)15-s + (0.613 − 0.316i)16-s + (3.97 + 3.79i)17-s + ⋯
L(s)  = 1  + (−1.25 − 0.984i)2-s + (−0.470 − 0.334i)3-s + (0.362 + 1.49i)4-s + (1.73 − 0.334i)5-s + (0.259 + 0.882i)6-s + (−0.897 + 0.440i)7-s + (0.356 − 0.779i)8-s + (0.109 + 0.315i)9-s + (−2.50 − 1.29i)10-s + (−0.281 − 0.357i)11-s + (0.330 − 0.824i)12-s + (−0.251 − 0.391i)13-s + (1.55 + 0.331i)14-s + (−0.929 − 0.424i)15-s + (0.153 − 0.0790i)16-s + (0.964 + 0.919i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.413 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.413 + 0.910i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (61, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.413 + 0.910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.413837 - 0.642718i\)
\(L(\frac12)\) \(\approx\) \(0.413837 - 0.642718i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.814 + 0.580i)T \)
7 \( 1 + (2.37 - 1.16i)T \)
23 \( 1 + (-3.85 + 2.85i)T \)
good2 \( 1 + (1.77 + 1.39i)T + (0.471 + 1.94i)T^{2} \)
5 \( 1 + (-3.88 + 0.748i)T + (4.64 - 1.85i)T^{2} \)
11 \( 1 + (0.933 + 1.18i)T + (-2.59 + 10.6i)T^{2} \)
13 \( 1 + (0.906 + 1.41i)T + (-5.40 + 11.8i)T^{2} \)
17 \( 1 + (-3.97 - 3.79i)T + (0.808 + 16.9i)T^{2} \)
19 \( 1 + (-4.08 + 3.89i)T + (0.904 - 18.9i)T^{2} \)
29 \( 1 + (0.560 - 0.164i)T + (24.3 - 15.6i)T^{2} \)
31 \( 1 + (0.900 + 9.43i)T + (-30.4 + 5.86i)T^{2} \)
37 \( 1 + (6.81 - 2.35i)T + (29.0 - 22.8i)T^{2} \)
41 \( 1 + (-4.74 + 4.11i)T + (5.83 - 40.5i)T^{2} \)
43 \( 1 + (5.34 - 2.44i)T + (28.1 - 32.4i)T^{2} \)
47 \( 1 + (-2.89 - 1.67i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-6.96 - 0.331i)T + (52.7 + 5.03i)T^{2} \)
59 \( 1 + (-5.62 + 10.9i)T + (-34.2 - 48.0i)T^{2} \)
61 \( 1 + (-3.21 - 4.51i)T + (-19.9 + 57.6i)T^{2} \)
67 \( 1 + (3.33 + 8.33i)T + (-48.4 + 46.2i)T^{2} \)
71 \( 1 + (1.14 - 7.95i)T + (-68.1 - 20.0i)T^{2} \)
73 \( 1 + (8.99 - 2.18i)T + (64.8 - 33.4i)T^{2} \)
79 \( 1 + (8.90 - 0.424i)T + (78.6 - 7.50i)T^{2} \)
83 \( 1 + (-2.70 + 3.12i)T + (-11.8 - 82.1i)T^{2} \)
89 \( 1 + (7.15 + 0.683i)T + (87.3 + 16.8i)T^{2} \)
97 \( 1 + (-6.38 - 7.36i)T + (-13.8 + 96.0i)T^{2} \)
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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

−10.37071147133495692779131721687, −9.912965012840771840292792752996, −9.216214648898416868383735195279, −8.430659416546744932696098267530, −7.13907544341941005785394790918, −5.93164275058698367882877479390, −5.36540691466708475612404498408, −3.06268790171591697243313078511, −2.15042223064638044732672292437, −0.847586873167542446049498799795, 1.32220010435998226890872358266, 3.17128707516677868843865387633, 5.30491054180059905918991394897, 5.80722068306433427609656971528, 6.94327771277384086288619251662, 7.24419310278002772820378288296, 8.905490936250982497218251341548, 9.563143981631387067075397565373, 10.10309084961626939741500664950, 10.50305299208065048813775705264

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