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Properties

Label 2-483-161.10-c1-0-7
Degree $2$
Conductor $483$
Sign $-0.447 - 0.894i$
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  + (−0.838 + 2.42i)2-s + (−0.458 − 0.888i)3-s + (−3.59 − 2.83i)4-s + (−2.60 + 0.248i)5-s + (2.53 − 0.365i)6-s + (2.49 − 0.872i)7-s + (5.56 − 3.57i)8-s + (−0.580 + 0.814i)9-s + (1.58 − 6.52i)10-s + (0.681 − 0.235i)11-s + (−0.866 + 4.49i)12-s + (0.235 + 0.803i)13-s + (0.0191 + 6.78i)14-s + (1.41 + 2.20i)15-s + (1.84 + 7.59i)16-s + (3.50 + 1.40i)17-s + ⋯
L(s)  = 1  + (−0.593 + 1.71i)2-s + (−0.264 − 0.513i)3-s + (−1.79 − 1.41i)4-s + (−1.16 + 0.111i)5-s + (1.03 − 0.149i)6-s + (0.944 − 0.329i)7-s + (1.96 − 1.26i)8-s + (−0.193 + 0.271i)9-s + (0.500 − 2.06i)10-s + (0.205 − 0.0711i)11-s + (−0.250 + 1.29i)12-s + (0.0654 + 0.222i)13-s + (0.00510 + 1.81i)14-s + (0.365 + 0.568i)15-s + (0.460 + 1.89i)16-s + (0.849 + 0.339i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.447 - 0.894i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (10, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.447 - 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.376297 + 0.608953i\)
\(L(\frac12)\) \(\approx\) \(0.376297 + 0.608953i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.458 + 0.888i)T \)
7 \( 1 + (-2.49 + 0.872i)T \)
23 \( 1 + (3.97 - 2.68i)T \)
good2 \( 1 + (0.838 - 2.42i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (2.60 - 0.248i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-0.681 + 0.235i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.235 - 0.803i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-3.50 - 1.40i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (-3.75 + 1.50i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-1.43 - 9.96i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (6.94 - 0.330i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-9.19 - 6.54i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-3.47 - 1.58i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-3.07 + 4.78i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-4.30 + 2.48i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-1.21 - 1.27i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (-5.63 - 1.36i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-6.55 - 3.37i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-1.46 - 7.61i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-2.47 - 2.85i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-3.05 + 3.88i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-3.60 + 3.77i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (5.84 + 12.8i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.185 + 3.88i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (1.24 - 2.71i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.28997060851891953606506463260, −10.21043951191363235637807557946, −9.000706816185552079760008812088, −8.186263370190812112493678975301, −7.50124201972226790696038101046, −7.12351796893659054461693810727, −5.85082847669556453223821218941, −5.00543994733378159690068458490, −3.85140959276898367676293389803, −1.08586197446275965174979459139, 0.74627391610420343913488657948, 2.42151737292430896878714349463, 3.78317126313968989160193079498, 4.32341636044549764723610311724, 5.58605765951999627222383126838, 7.79010409464433911603529399805, 8.067998755249217498734566673530, 9.267356479559423729969959722044, 9.895755272062467825664784672053, 10.99807485209662347115430789655

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