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Properties

Label 2-483-21.20-c1-0-59
Degree $2$
Conductor $483$
Sign $0.691 - 0.722i$
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  − 2.52i·2-s + (−0.292 − 1.70i)3-s − 4.39·4-s − 0.707·5-s + (−4.31 + 0.738i)6-s + (−2.12 + 1.57i)7-s + 6.06i·8-s + (−2.82 + 0.997i)9-s + 1.78i·10-s − 5.52i·11-s + (1.28 + 7.50i)12-s − 1.55i·13-s + (3.98 + 5.37i)14-s + (0.206 + 1.20i)15-s + 6.54·16-s + 4.61·17-s + ⋯
L(s)  = 1  − 1.78i·2-s + (−0.168 − 0.985i)3-s − 2.19·4-s − 0.316·5-s + (−1.76 + 0.301i)6-s + (−0.803 + 0.595i)7-s + 2.14i·8-s + (−0.943 + 0.332i)9-s + 0.565i·10-s − 1.66i·11-s + (0.370 + 2.16i)12-s − 0.430i·13-s + (1.06 + 1.43i)14-s + (0.0533 + 0.311i)15-s + 1.63·16-s + 1.12·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.691 - 0.722i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $0.691 - 0.722i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (461, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ 0.691 - 0.722i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.327934 + 0.140138i\)
\(L(\frac12)\) \(\approx\) \(0.327934 + 0.140138i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.292 + 1.70i)T \)
7 \( 1 + (2.12 - 1.57i)T \)
23 \( 1 - iT \)
good2 \( 1 + 2.52iT - 2T^{2} \)
5 \( 1 + 0.707T + 5T^{2} \)
11 \( 1 + 5.52iT - 11T^{2} \)
13 \( 1 + 1.55iT - 13T^{2} \)
17 \( 1 - 4.61T + 17T^{2} \)
19 \( 1 - 7.93iT - 19T^{2} \)
29 \( 1 + 2.49iT - 29T^{2} \)
31 \( 1 + 5.71iT - 31T^{2} \)
37 \( 1 + 0.902T + 37T^{2} \)
41 \( 1 + 3.75T + 41T^{2} \)
43 \( 1 + 9.08T + 43T^{2} \)
47 \( 1 + 9.85T + 47T^{2} \)
53 \( 1 + 5.71iT - 53T^{2} \)
59 \( 1 + 12.0T + 59T^{2} \)
61 \( 1 - 9.47iT - 61T^{2} \)
67 \( 1 + 0.757T + 67T^{2} \)
71 \( 1 + 2.66iT - 71T^{2} \)
73 \( 1 + 1.77iT - 73T^{2} \)
79 \( 1 - 6.58T + 79T^{2} \)
83 \( 1 + 7.55T + 83T^{2} \)
89 \( 1 - 12.1T + 89T^{2} \)
97 \( 1 + 0.965iT - 97T^{2} \)
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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

−10.35714224121986136729887417669, −9.624430073973994278611987264892, −8.433200188978650194956974447937, −7.947527270522312961051939765515, −6.13225796484883264781356943935, −5.50529546999246004432708593051, −3.57290768766780584017204538093, −3.08746397082505289743180741196, −1.65825255606006348028393150801, −0.22388504402013292405308831745, 3.44247476454748190116756645149, 4.58254162102019847831238394763, 5.06971049112017982303019181695, 6.44903855477155637899606015769, 7.02885695915943375853503186577, 7.942106072624354042852880934429, 9.127127433066292803589798231620, 9.640923231010850998415421592017, 10.43672886066633621425499508728, 11.81010267670949510375518287928

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