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Properties

Label 2-483-161.10-c1-0-8
Degree $2$
Conductor $483$
Sign $0.422 - 0.906i$
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  + (−0.0815 + 0.235i)2-s + (−0.458 − 0.888i)3-s + (1.52 + 1.19i)4-s + (−1.53 + 0.146i)5-s + (0.246 − 0.0354i)6-s + (−2.59 − 0.498i)7-s + (−0.826 + 0.531i)8-s + (−0.580 + 0.814i)9-s + (0.0907 − 0.374i)10-s + (3.95 − 1.36i)11-s + (0.366 − 1.90i)12-s + (1.89 + 6.46i)13-s + (0.329 − 0.571i)14-s + (0.834 + 1.29i)15-s + (0.855 + 3.52i)16-s + (7.55 + 3.02i)17-s + ⋯
L(s)  = 1  + (−0.0576 + 0.166i)2-s + (−0.264 − 0.513i)3-s + (0.761 + 0.598i)4-s + (−0.687 + 0.0656i)5-s + (0.100 − 0.0144i)6-s + (−0.982 − 0.188i)7-s + (−0.292 + 0.187i)8-s + (−0.193 + 0.271i)9-s + (0.0287 − 0.118i)10-s + (1.19 − 0.412i)11-s + (0.105 − 0.549i)12-s + (0.526 + 1.79i)13-s + (0.0880 − 0.152i)14-s + (0.215 + 0.335i)15-s + (0.213 + 0.882i)16-s + (1.83 + 0.733i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.422 - 0.906i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.422 - 0.906i$
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: \((2,\ 483,\ (\ :1/2),\ 0.422 - 0.906i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.00938 + 0.642879i\)
\(L(\frac12)\) \(\approx\) \(1.00938 + 0.642879i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.59 + 0.498i)T \)
23 \( 1 + (-4.68 + 1.04i)T \)
good2 \( 1 + (0.0815 - 0.235i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (1.53 - 0.146i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (-3.95 + 1.36i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-1.89 - 6.46i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-7.55 - 3.02i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (5.56 - 2.22i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (-0.253 - 1.76i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (2.00 - 0.0956i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-6.53 - 4.65i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (10.9 + 5.00i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.41 + 3.75i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (1.54 - 0.893i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-4.13 - 4.33i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.66 + 1.13i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-0.0486 - 0.0250i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (0.194 + 1.00i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-0.0712 - 0.0822i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (-6.78 + 8.62i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-5.03 + 5.27i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (2.15 + 4.72i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (-0.166 + 3.48i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (2.05 - 4.49i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.38137057596693655122492944288, −10.46098945769792735850866691359, −9.145805562955084355319511386854, −8.339392641870904321656646283329, −7.31486519094770889643545346085, −6.56035236788747655314258596554, −6.06576086551258042004098859896, −4.00624886762147973528035338275, −3.38715958541227183236507620887, −1.62762143785326746678683802096, 0.812468329020902047045489875025, 2.92188215639017601493719608475, 3.75583116366367743922741955368, 5.28345262973948212002122806019, 6.11578631805498107421099813452, 7.01099457697410308229712600016, 8.094418598899613366821684310723, 9.391260309518861243751338640900, 9.965788504861359711973754311976, 10.80254740037124396418676808938

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