↑

Properties

Label 2-483-21.20-c1-0-54
Degree $2$
Conductor $483$
Sign $-0.720 + 0.693i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 1.64i·2-s + (1.53 − 0.796i)3-s − 0.694·4-s + 0.500·5-s + (−1.30 − 2.52i)6-s + (−0.751 − 2.53i)7-s − 2.14i·8-s + (1.73 − 2.44i)9-s − 0.821i·10-s − 0.0158i·11-s + (−1.06 + 0.552i)12-s + 5.86i·13-s + (−4.16 + 1.23i)14-s + (0.770 − 0.398i)15-s − 4.90·16-s + 3.84·17-s + ⋯
L(s)  = 1  − 1.16i·2-s + (0.888 − 0.459i)3-s − 0.347·4-s + 0.223·5-s + (−0.533 − 1.03i)6-s + (−0.284 − 0.958i)7-s − 0.757i·8-s + (0.577 − 0.816i)9-s − 0.259i·10-s − 0.00477i·11-s + (−0.308 + 0.159i)12-s + 1.62i·13-s + (−1.11 + 0.329i)14-s + (0.198 − 0.102i)15-s − 1.22·16-s + 0.933·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.720 + 0.693i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.720 + 0.693i$
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: \((2,\ 483,\ (\ :1/2),\ -0.720 + 0.693i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.760726 - 1.88884i\)
\(L(\frac12)\) \(\approx\) \(0.760726 - 1.88884i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-1.53 + 0.796i)T \)
7 \( 1 + (0.751 + 2.53i)T \)
23 \( 1 - iT \)
good2 \( 1 + 1.64iT - 2T^{2} \)
5 \( 1 - 0.500T + 5T^{2} \)
11 \( 1 + 0.0158iT - 11T^{2} \)
13 \( 1 - 5.86iT - 13T^{2} \)
17 \( 1 - 3.84T + 17T^{2} \)
19 \( 1 - 4.50iT - 19T^{2} \)
29 \( 1 + 7.34iT - 29T^{2} \)
31 \( 1 - 1.96iT - 31T^{2} \)
37 \( 1 - 0.934T + 37T^{2} \)
41 \( 1 + 5.56T + 41T^{2} \)
43 \( 1 + 0.158T + 43T^{2} \)
47 \( 1 - 9.11T + 47T^{2} \)
53 \( 1 - 6.54iT - 53T^{2} \)
59 \( 1 - 3.35T + 59T^{2} \)
61 \( 1 - 10.3iT - 61T^{2} \)
67 \( 1 - 10.0T + 67T^{2} \)
71 \( 1 + 2.44iT - 71T^{2} \)
73 \( 1 + 4.33iT - 73T^{2} \)
79 \( 1 - 15.1T + 79T^{2} \)
83 \( 1 + 7.61T + 83T^{2} \)
89 \( 1 - 6.59T + 89T^{2} \)
97 \( 1 + 3.68iT - 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.51229611275430561255796675259, −9.816964022050501091144725540738, −9.249624266928071763739443527263, −7.928136842641161263892416471275, −7.09508494611930497089038316112, −6.20142596411583002823584133021, −4.15451693268044597051648241459, −3.61528174360744290114731091413, −2.27329492219623319354195637877, −1.27598869307896192194603878566, 2.35845499292191678015352346108, 3.34976730826761529259288904765, 5.10101879710846574086703116242, 5.60204217842991276904562031514, 6.82077848779002813890285384558, 7.84057931386535151505953708765, 8.417209954005007843193164879673, 9.282892029373205850080875610779, 10.15680948987407381485312002736, 11.15849262705317248517704871999

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