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Properties

Label 2-722-19.18-c2-0-9
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

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41i·2-s − 3.88i·3-s − 2.00·4-s − 8.89·5-s + 5.48·6-s − 5.10·7-s − 2.82i·8-s − 6.06·9-s − 12.5i·10-s − 5.40·11-s + 7.76i·12-s − 12.5i·13-s − 7.22i·14-s + 34.5i·15-s + 4.00·16-s − 25.7·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 1.29i·3-s − 0.500·4-s − 1.77·5-s + 0.914·6-s − 0.729·7-s − 0.353i·8-s − 0.674·9-s − 1.25i·10-s − 0.491·11-s + 0.646i·12-s − 0.963i·13-s − 0.515i·14-s + 2.30i·15-s + 0.250·16-s − 1.51·17-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

\(L(\frac{3}{2})\) \(\approx\) \(0.4063735690\)
\(L(\frac12)\) \(\approx\) \(0.4063735690\)
\(L(2)\) 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 \( 1 + 3.88iT - 9T^{2} \)
5 \( 1 + 8.89T + 25T^{2} \)
7 \( 1 + 5.10T + 49T^{2} \)
11 \( 1 + 5.40T + 121T^{2} \)
13 \( 1 + 12.5iT - 169T^{2} \)
17 \( 1 + 25.7T + 289T^{2} \)
23 \( 1 - 8.37T + 529T^{2} \)
29 \( 1 - 47.2iT - 841T^{2} \)
31 \( 1 + 22.0iT - 961T^{2} \)
37 \( 1 - 16.7iT - 1.36e3T^{2} \)
41 \( 1 - 5.08iT - 1.68e3T^{2} \)
43 \( 1 - 37.6T + 1.84e3T^{2} \)
47 \( 1 - 44.9T + 2.20e3T^{2} \)
53 \( 1 - 28.2iT - 2.80e3T^{2} \)
59 \( 1 + 42.1iT - 3.48e3T^{2} \)
61 \( 1 - 49.5T + 3.72e3T^{2} \)
67 \( 1 - 25.8iT - 4.48e3T^{2} \)
71 \( 1 - 59.9iT - 5.04e3T^{2} \)
73 \( 1 - 34.0T + 5.32e3T^{2} \)
79 \( 1 + 73.0iT - 6.24e3T^{2} \)
83 \( 1 - 71.0T + 6.88e3T^{2} \)
89 \( 1 + 52.3iT - 7.92e3T^{2} \)
97 \( 1 - 43.4iT - 9.40e3T^{2} \)
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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

−10.49720344841824963282138873239, −9.037099762689534937397304313396, −8.272982023070389463574017809838, −7.59094412686739955669798457596, −7.04145616153408863831738473387, −6.30641451139747598955878446389, −4.99356677035284675198624292884, −3.87004689159644145984970820383, −2.76005794005653536922017170934, −0.72322946197040830941250995296, 0.23193532455969028464761203753, 2.62881992822487436680696596201, 3.79622099463534111963161472757, 4.17429604564245643944678899758, 4.98611073908605592269158517635, 6.59117332840220936765980408802, 7.62107202020026830805177793271, 8.716159316841446120907337913915, 9.240900092717927121742503044029, 10.20003414996507997991781462952

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