↑

Properties

Label 2-722-19.18-c2-0-3
Degree $2$
Conductor $722$
Sign $-0.506 - 0.862i$
Analytic cond. $19.6730$
Root an. cond. $4.43543$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 5.31i·3-s − 2.00·4-s − 1.38·5-s + 7.51·6-s + 8.79·7-s − 2.82i·8-s − 19.2·9-s − 1.96i·10-s − 12.1·11-s + 10.6i·12-s + 16.8i·13-s + 12.4i·14-s + 7.38i·15-s + 4.00·16-s − 15.8·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.77i·3-s − 0.500·4-s − 0.277·5-s + 1.25·6-s + 1.25·7-s − 0.353i·8-s − 2.13·9-s − 0.196i·10-s − 1.10·11-s + 0.885i·12-s + 1.29i·13-s + 0.888i·14-s + 0.492i·15-s + 0.250·16-s − 0.935·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.506 - 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.506 - 0.862i$
Analytic conductor: \(19.6730\)
Root analytic conductor: \(4.43543\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1),\ -0.506 - 0.862i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.3660765497\)
\(L(\frac12)\) \(\approx\) \(0.3660765497\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41iT \)
19 \( 1 \)
good3 \( 1 + 5.31iT - 9T^{2} \)
5 \( 1 + 1.38T + 25T^{2} \)
7 \( 1 - 8.79T + 49T^{2} \)
11 \( 1 + 12.1T + 121T^{2} \)
13 \( 1 - 16.8iT - 169T^{2} \)
17 \( 1 + 15.8T + 289T^{2} \)
23 \( 1 + 6.61T + 529T^{2} \)
29 \( 1 - 20.1iT - 841T^{2} \)
31 \( 1 - 13.6iT - 961T^{2} \)
37 \( 1 + 42.6iT - 1.36e3T^{2} \)
41 \( 1 - 24.9iT - 1.68e3T^{2} \)
43 \( 1 - 39.2T + 1.84e3T^{2} \)
47 \( 1 + 18.4T + 2.20e3T^{2} \)
53 \( 1 - 20.6iT - 2.80e3T^{2} \)
59 \( 1 - 98.6iT - 3.48e3T^{2} \)
61 \( 1 + 94.2T + 3.72e3T^{2} \)
67 \( 1 + 31.7iT - 4.48e3T^{2} \)
71 \( 1 - 62.0iT - 5.04e3T^{2} \)
73 \( 1 - 83.1T + 5.32e3T^{2} \)
79 \( 1 - 43.0iT - 6.24e3T^{2} \)
83 \( 1 + 21.9T + 6.88e3T^{2} \)
89 \( 1 - 3.56iT - 7.92e3T^{2} \)
97 \( 1 + 54.5iT - 9.40e3T^{2} \)
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

−10.78896546897072807149749430503, −9.148567872162220148634840704774, −8.383225076410818620482491468618, −7.71230122437096347609072353058, −7.19556839609252343071995421360, −6.29224294356273835785004260690, −5.35057395342824684198296164761, −4.30395730457184682384849070453, −2.43065351409595822897861201857, −1.48572579227904839207553911403, 0.12520226511332010128478363330, 2.32132208736687643255437769571, 3.40211775308987474001988420305, 4.41826986799601120431787183943, 4.99841799903105367467484584197, 5.79360747569263595432152064556, 7.940583487845082427748534509211, 8.223734732830319943928149190929, 9.342895131148983228812327335226, 10.11939361360822994301754597408

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