↑

Properties

Label 2-483-161.160-c1-0-29
Degree $2$
Conductor $483$
Sign $-0.822 + 0.568i$
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  + 1.17·2-s + i·3-s − 0.611·4-s − 1.67·5-s + 1.17i·6-s + (−2.04 − 1.67i)7-s − 3.07·8-s − 9-s − 1.97·10-s − 3.72i·11-s − 0.611i·12-s + 0.821i·13-s + (−2.40 − 1.97i)14-s − 1.67i·15-s − 2.40·16-s − 1.97·17-s + ⋯
L(s)  = 1  + 0.833·2-s + 0.577i·3-s − 0.305·4-s − 0.751·5-s + 0.480i·6-s + (−0.772 − 0.634i)7-s − 1.08·8-s − 0.333·9-s − 0.625·10-s − 1.12i·11-s − 0.176i·12-s + 0.227i·13-s + (−0.643 − 0.528i)14-s − 0.433i·15-s − 0.600·16-s − 0.480·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.822 + 0.568i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(483\)    =    \(3 \cdot 7 \cdot 23\)
Sign: $-0.822 + 0.568i$
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: \((2,\ 483,\ (\ :1/2),\ -0.822 + 0.568i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0799146 - 0.256064i\)
\(L(\frac12)\) \(\approx\) \(0.0799146 - 0.256064i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.04 + 1.67i)T \)
23 \( 1 + (-4.61 - 1.31i)T \)
good2 \( 1 - 1.17T + 2T^{2} \)
5 \( 1 + 1.67T + 5T^{2} \)
11 \( 1 + 3.72iT - 11T^{2} \)
13 \( 1 - 0.821iT - 13T^{2} \)
17 \( 1 + 1.97T + 17T^{2} \)
19 \( 1 + 3.72T + 19T^{2} \)
29 \( 1 + 5.79T + 29T^{2} \)
31 \( 1 + 0.566iT - 31T^{2} \)
37 \( 1 + 1.25iT - 37T^{2} \)
41 \( 1 - 5iT - 41T^{2} \)
43 \( 1 - 7.01iT - 43T^{2} \)
47 \( 1 + 5.14iT - 47T^{2} \)
53 \( 1 - 0.0649iT - 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 9.36T + 61T^{2} \)
67 \( 1 - 5.03iT - 67T^{2} \)
71 \( 1 + 6.40T + 71T^{2} \)
73 \( 1 + 15.8iT - 73T^{2} \)
79 \( 1 - 17.6iT - 79T^{2} \)
83 \( 1 - 9.29T + 83T^{2} \)
89 \( 1 - 11.8T + 89T^{2} \)
97 \( 1 + 14.9T + 97T^{2} \)
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

−10.88771595356992649746972702691, −9.650941817396785605366911651707, −8.942696043411943943744568862544, −7.980566075188895821209219385468, −6.66283133235306885942281131100, −5.77909568178462320244785703429, −4.59175400306586498058978143752, −3.81321308212743259151981120108, −3.10848759635813834368698478809, −0.11944497634423176449690490224, 2.39689420420267530762264374932, 3.60781188683995643858375371487, 4.59366108972420318057126630811, 5.67750466629620104268835873245, 6.63326377697687900315122418344, 7.54860818422146094569293278695, 8.738436810707663188732560250198, 9.341203085426113979667500933583, 10.59359609633889289755666632553, 11.81456975317058832204635435349

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