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Properties

Label 2-483-161.25-c1-0-12
Degree $2$
Conductor $483$
Sign $0.929 - 0.367i$
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  + (0.0438 − 0.920i)2-s + (0.928 + 0.371i)3-s + (1.14 + 0.109i)4-s + (0.838 + 3.45i)5-s + (0.382 − 0.838i)6-s + (−0.905 + 2.48i)7-s + (0.413 − 2.87i)8-s + (0.723 + 0.690i)9-s + (3.21 − 0.620i)10-s + (−0.0892 − 1.87i)11-s + (1.02 + 0.527i)12-s + (−4.13 − 4.77i)13-s + (2.24 + 0.942i)14-s + (−0.506 + 3.52i)15-s + (−0.365 − 0.0705i)16-s + (4.02 + 5.65i)17-s + ⋯
L(s)  = 1  + (0.0309 − 0.650i)2-s + (0.535 + 0.214i)3-s + (0.572 + 0.0547i)4-s + (0.375 + 1.54i)5-s + (0.156 − 0.342i)6-s + (−0.342 + 0.939i)7-s + (0.146 − 1.01i)8-s + (0.241 + 0.230i)9-s + (1.01 − 0.196i)10-s + (−0.0269 − 0.564i)11-s + (0.295 + 0.152i)12-s + (−1.14 − 1.32i)13-s + (0.600 + 0.251i)14-s + (−0.130 + 0.909i)15-s + (−0.0914 − 0.0176i)16-s + (0.975 + 1.37i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.929 - 0.367i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.929 - 0.367i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.929 - 0.367i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00506 + 0.381957i\)
\(L(\frac12)\) \(\approx\) \(2.00506 + 0.381957i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.928 - 0.371i)T \)
7 \( 1 + (0.905 - 2.48i)T \)
23 \( 1 + (-4.42 - 1.83i)T \)
good2 \( 1 + (-0.0438 + 0.920i)T + (-1.99 - 0.190i)T^{2} \)
5 \( 1 + (-0.838 - 3.45i)T + (-4.44 + 2.29i)T^{2} \)
11 \( 1 + (0.0892 + 1.87i)T + (-10.9 + 1.04i)T^{2} \)
13 \( 1 + (4.13 + 4.77i)T + (-1.85 + 12.8i)T^{2} \)
17 \( 1 + (-4.02 - 5.65i)T + (-5.56 + 16.0i)T^{2} \)
19 \( 1 + (1.94 - 2.72i)T + (-6.21 - 17.9i)T^{2} \)
29 \( 1 + (-2.46 + 5.39i)T + (-18.9 - 21.9i)T^{2} \)
31 \( 1 + (0.402 + 0.316i)T + (7.30 + 30.1i)T^{2} \)
37 \( 1 + (-2.37 - 2.25i)T + (1.76 + 36.9i)T^{2} \)
41 \( 1 + (-1.00 - 0.294i)T + (34.4 + 22.1i)T^{2} \)
43 \( 1 + (1.52 + 10.6i)T + (-41.2 + 12.1i)T^{2} \)
47 \( 1 + (1.36 + 2.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (1.92 + 5.55i)T + (-41.6 + 32.7i)T^{2} \)
59 \( 1 + (0.121 - 0.0233i)T + (54.7 - 21.9i)T^{2} \)
61 \( 1 + (2.10 - 0.842i)T + (44.1 - 42.0i)T^{2} \)
67 \( 1 + (9.30 - 4.79i)T + (38.8 - 54.5i)T^{2} \)
71 \( 1 + (-4.06 - 2.61i)T + (29.4 + 64.5i)T^{2} \)
73 \( 1 + (-8.50 - 0.812i)T + (71.6 + 13.8i)T^{2} \)
79 \( 1 + (-4.69 + 13.5i)T + (-62.0 - 48.8i)T^{2} \)
83 \( 1 + (3.15 - 0.925i)T + (69.8 - 44.8i)T^{2} \)
89 \( 1 + (-13.6 + 10.7i)T + (20.9 - 86.4i)T^{2} \)
97 \( 1 + (10.8 + 3.18i)T + (81.6 + 52.4i)T^{2} \)
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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

−10.78870760185051609255436792441, −10.24787325898033782544274472097, −9.732642129189399735418154053100, −8.275738263040455003263715554937, −7.42869577087492890726585112071, −6.35896534413868113773913079237, −5.61066609074191736926170118642, −3.52116640550760839512796972086, −2.96303286772120182299892729578, −2.12960042664402825172872862508, 1.29673937470555141807509538093, 2.67890959433436899617536739017, 4.56513240502286255020798960610, 5.06614588408538631945081827534, 6.61018041913715337013407334032, 7.22372265304442017908270689863, 8.028505892351951105917844685329, 9.282497858821674322686887614720, 9.561197088954464381311013998157, 10.92096732913862509745553144402

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