↑

Properties

Label 2-483-161.160-c1-0-21
Degree $2$
Conductor $483$
Sign $-0.459 + 0.887i$
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  − 2.69·2-s − i·3-s + 5.27·4-s + 2.58·5-s + 2.69i·6-s + (0.551 − 2.58i)7-s − 8.81·8-s − 9-s − 6.97·10-s − 3.13i·11-s − 5.27i·12-s − 4.69i·13-s + (−1.48 + 6.97i)14-s − 2.58i·15-s + 13.2·16-s − 6.97·17-s + ⋯
L(s)  = 1  − 1.90·2-s − 0.577i·3-s + 2.63·4-s + 1.15·5-s + 1.10i·6-s + (0.208 − 0.978i)7-s − 3.11·8-s − 0.333·9-s − 2.20·10-s − 0.946i·11-s − 1.52i·12-s − 1.30i·13-s + (−0.397 + 1.86i)14-s − 0.668i·15-s + 3.30·16-s − 1.69·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.459 + 0.887i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.459 + 0.887i$
Analytic conductor: \(3.85677\)
Root analytic conductor: \(1.96386\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{483} (160, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 483,\ (\ :1/2),\ -0.459 + 0.887i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.333434 - 0.548218i\)
\(L(\frac12)\) \(\approx\) \(0.333434 - 0.548218i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
7 \( 1 + (-0.551 + 2.58i)T \)
23 \( 1 + (1.27 - 4.62i)T \)
good2 \( 1 + 2.69T + 2T^{2} \)
5 \( 1 - 2.58T + 5T^{2} \)
11 \( 1 + 3.13iT - 11T^{2} \)
13 \( 1 + 4.69iT - 13T^{2} \)
17 \( 1 + 6.97T + 17T^{2} \)
19 \( 1 - 3.13T + 19T^{2} \)
29 \( 1 - 3.96T + 29T^{2} \)
31 \( 1 - 2.57iT - 31T^{2} \)
37 \( 1 - 2.90iT - 37T^{2} \)
41 \( 1 + 5iT - 41T^{2} \)
43 \( 1 - 0.785iT - 43T^{2} \)
47 \( 1 + 12.3iT - 47T^{2} \)
53 \( 1 - 7.52iT - 53T^{2} \)
59 \( 1 - 3.15iT - 59T^{2} \)
61 \( 1 + 8.22T + 61T^{2} \)
67 \( 1 - 7.76iT - 67T^{2} \)
71 \( 1 - 9.23T + 71T^{2} \)
73 \( 1 + 9.65iT - 73T^{2} \)
79 \( 1 - 1.50iT - 79T^{2} \)
83 \( 1 - 15.7T + 83T^{2} \)
89 \( 1 - 2.18T + 89T^{2} \)
97 \( 1 + 7.74T + 97T^{2} \)
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

−10.55289162126272631724848404730, −9.786875046792406864952671579111, −8.867022423123148520839922880180, −8.123833725587814512535408459687, −7.25423280727960745724749017432, −6.44579436932572117904141437746, −5.55441261631829241597222361657, −3.08992512950987519187376335928, −1.85077960904515098745878686299, −0.69374506625282558242126893483, 1.89767652545717616058405754272, 2.47194957969292552127991902834, 4.73820968232305583255398441794, 6.18147012559316350330245481780, 6.69489634799951650074855955396, 8.020166741315231600809367152531, 9.081816601024755407555201253495, 9.309458757608504529381928822044, 10.01331305405914237442085857887, 10.97882201834483409541273938562

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