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Properties

Label 2-722-19.18-c2-0-45
Degree $2$
Conductor $722$
Sign $0.987 + 0.159i$
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 − 1.64i·3-s − 2.00·4-s + 4.64·5-s + 2.33·6-s + 4.36·7-s − 2.82i·8-s + 6.27·9-s + 6.56i·10-s + 5.73·11-s + 3.29i·12-s − 18.6i·13-s + 6.16i·14-s − 7.65i·15-s + 4.00·16-s + 6.59·17-s + ⋯
L(s)  = 1  + 0.707i·2-s − 0.549i·3-s − 0.500·4-s + 0.928·5-s + 0.388·6-s + 0.623·7-s − 0.353i·8-s + 0.697·9-s + 0.656i·10-s + 0.521·11-s + 0.274i·12-s − 1.43i·13-s + 0.440i·14-s − 0.510i·15-s + 0.250·16-s + 0.387·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.987 + 0.159i$
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: \((2,\ 722,\ (\ :1),\ 0.987 + 0.159i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(2.405982057\)
\(L(\frac12)\) \(\approx\) \(2.405982057\)
\(L(2)\) 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 \( 1 + 1.64iT - 9T^{2} \)
5 \( 1 - 4.64T + 25T^{2} \)
7 \( 1 - 4.36T + 49T^{2} \)
11 \( 1 - 5.73T + 121T^{2} \)
13 \( 1 + 18.6iT - 169T^{2} \)
17 \( 1 - 6.59T + 289T^{2} \)
23 \( 1 + 25.3T + 529T^{2} \)
29 \( 1 + 3.54iT - 841T^{2} \)
31 \( 1 - 29.6iT - 961T^{2} \)
37 \( 1 + 62.3iT - 1.36e3T^{2} \)
41 \( 1 - 64.7iT - 1.68e3T^{2} \)
43 \( 1 - 82.6T + 1.84e3T^{2} \)
47 \( 1 + 12.1T + 2.20e3T^{2} \)
53 \( 1 + 77.7iT - 2.80e3T^{2} \)
59 \( 1 - 18.7iT - 3.48e3T^{2} \)
61 \( 1 - 70.1T + 3.72e3T^{2} \)
67 \( 1 + 44.6iT - 4.48e3T^{2} \)
71 \( 1 + 28.1iT - 5.04e3T^{2} \)
73 \( 1 - 59.1T + 5.32e3T^{2} \)
79 \( 1 - 30.7iT - 6.24e3T^{2} \)
83 \( 1 - 162.T + 6.88e3T^{2} \)
89 \( 1 + 18.2iT - 7.92e3T^{2} \)
97 \( 1 - 134. iT - 9.40e3T^{2} \)
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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

−10.00985321417191881237375679110, −9.348526462608867582119336520299, −8.115549446782243099211249580243, −7.68553885921319569472740060333, −6.59082605883626979756852392424, −5.84809829574888247895977316106, −5.02458689032755007885735845746, −3.78109062466385344922050327274, −2.14925933789539914302453862383, −0.974586619920943700305591945918, 1.41446036047971390618383582767, 2.22498662173201565886747361185, 3.88132427973376483843694435089, 4.47643482699212880701116244791, 5.57643668457888339380797193432, 6.58761048346599132532167465452, 7.75002049080686878527270703721, 8.926115042921463718176305648118, 9.573604782324161494682332952972, 10.09359435220650823815132437845

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