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Properties

Label 2-483-161.10-c1-0-5
Degree $2$
Conductor $483$
Sign $-0.731 + 0.681i$
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  + (−0.632 + 1.82i)2-s + (0.458 + 0.888i)3-s + (−1.36 − 1.07i)4-s + (0.843 − 0.0805i)5-s + (−1.91 + 0.275i)6-s + (−2.38 + 1.15i)7-s + (−0.422 + 0.271i)8-s + (−0.580 + 0.814i)9-s + (−0.386 + 1.59i)10-s + (−1.15 + 0.399i)11-s + (0.329 − 1.70i)12-s + (0.0600 + 0.204i)13-s + (−0.595 − 5.08i)14-s + (0.458 + 0.712i)15-s + (−1.04 − 4.32i)16-s + (1.79 + 0.718i)17-s + ⋯
L(s)  = 1  + (−0.447 + 1.29i)2-s + (0.264 + 0.513i)3-s + (−0.683 − 0.537i)4-s + (0.377 − 0.0360i)5-s + (−0.781 + 0.112i)6-s + (−0.900 + 0.434i)7-s + (−0.149 + 0.0960i)8-s + (−0.193 + 0.271i)9-s + (−0.122 + 0.503i)10-s + (−0.348 + 0.120i)11-s + (0.0950 − 0.493i)12-s + (0.0166 + 0.0566i)13-s + (−0.159 − 1.35i)14-s + (0.118 + 0.184i)15-s + (−0.262 − 1.08i)16-s + (0.435 + 0.174i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.731 + 0.681i$
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: \((2,\ 483,\ (\ :1/2),\ -0.731 + 0.681i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.282546 - 0.717825i\)
\(L(\frac12)\) \(\approx\) \(0.282546 - 0.717825i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.38 - 1.15i)T \)
23 \( 1 + (-3.32 - 3.45i)T \)
good2 \( 1 + (0.632 - 1.82i)T + (-1.57 - 1.23i)T^{2} \)
5 \( 1 + (-0.843 + 0.0805i)T + (4.90 - 0.946i)T^{2} \)
11 \( 1 + (1.15 - 0.399i)T + (8.64 - 6.79i)T^{2} \)
13 \( 1 + (-0.0600 - 0.204i)T + (-10.9 + 7.02i)T^{2} \)
17 \( 1 + (-1.79 - 0.718i)T + (12.3 + 11.7i)T^{2} \)
19 \( 1 + (5.96 - 2.38i)T + (13.7 - 13.1i)T^{2} \)
29 \( 1 + (0.0808 + 0.562i)T + (-27.8 + 8.17i)T^{2} \)
31 \( 1 + (8.09 - 0.385i)T + (30.8 - 2.94i)T^{2} \)
37 \( 1 + (-3.80 - 2.70i)T + (12.1 + 34.9i)T^{2} \)
41 \( 1 + (-7.47 - 3.41i)T + (26.8 + 30.9i)T^{2} \)
43 \( 1 + (-2.30 + 3.58i)T + (-17.8 - 39.1i)T^{2} \)
47 \( 1 + (-4.76 + 2.75i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.58 + 2.70i)T + (-2.52 + 52.9i)T^{2} \)
59 \( 1 + (4.28 + 1.04i)T + (52.4 + 27.0i)T^{2} \)
61 \( 1 + (-9.05 - 4.66i)T + (35.3 + 49.6i)T^{2} \)
67 \( 1 + (-2.59 - 13.4i)T + (-62.2 + 24.9i)T^{2} \)
71 \( 1 + (-5.77 - 6.66i)T + (-10.1 + 70.2i)T^{2} \)
73 \( 1 + (3.86 - 4.92i)T + (-17.2 - 70.9i)T^{2} \)
79 \( 1 + (-1.07 + 1.12i)T + (-3.75 - 78.9i)T^{2} \)
83 \( 1 + (-1.23 - 2.70i)T + (-54.3 + 62.7i)T^{2} \)
89 \( 1 + (0.00527 - 0.110i)T + (-88.5 - 8.45i)T^{2} \)
97 \( 1 + (-4.32 + 9.47i)T + (-63.5 - 73.3i)T^{2} \)
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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

−11.38857900821064284539936327484, −10.22297241209329306248093991659, −9.484965493017961692037633017703, −8.834357028738201935614493623571, −7.932834604039011051805875245157, −6.98698059358984555857844181492, −5.96451956869033761842951782657, −5.41343903288796562233984795921, −3.86276447774748831194718069892, −2.48613139493014994953117929248, 0.49259801561913256373362953689, 2.09675994956299546696032010999, 3.00253127527382143800923877121, 4.12248444498020672900274652143, 5.89287101040300965573468005269, 6.74746576440106364328950554882, 7.87831896493619979191544171674, 9.099508795803101110702919849373, 9.517170347529855959184296612907, 10.66917867189844823466598777099

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