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Properties

Label 2-722-19.11-c1-0-22
Degree $2$
Conductor $722$
Sign $-0.813 + 0.582i$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (1.32 − 2.29i)3-s + (−0.499 − 0.866i)4-s + (0.822 − 1.42i)5-s + (−1.32 − 2.29i)6-s + 3.64·7-s − 0.999·8-s + (−2 − 3.46i)9-s + (−0.822 − 1.42i)10-s − 4.64·11-s − 2.64·12-s + (1 + 1.73i)13-s + (1.82 − 3.15i)14-s + (−2.17 − 3.77i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (0.763 − 1.32i)3-s + (−0.249 − 0.433i)4-s + (0.368 − 0.637i)5-s + (−0.540 − 0.935i)6-s + 1.37·7-s − 0.353·8-s + (−0.666 − 1.15i)9-s + (−0.260 − 0.450i)10-s − 1.40·11-s − 0.763·12-s + (0.277 + 0.480i)13-s + (0.487 − 0.843i)14-s + (−0.562 − 0.973i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.813 + 0.582i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (429, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.783711 - 2.44033i\)
\(L(\frac12)\) \(\approx\) \(0.783711 - 2.44033i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 + 0.866i)T \)
19 \( 1 \)
good3 \( 1 + (-1.32 + 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.822 + 1.42i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 3.64T + 7T^{2} \)
11 \( 1 + 4.64T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-0.822 - 1.42i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.822 - 1.42i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5.64T + 31T^{2} \)
37 \( 1 + 0.354T + 37T^{2} \)
41 \( 1 + (0.145 - 0.252i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.64 - 9.77i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.17 + 3.77i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.29 + 10.8i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.96 - 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.468 + 0.811i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.322 - 0.559i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (1.35 - 2.34i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (0.854 - 1.47i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.85 - 3.21i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.12284083215114286165492193808, −8.992408892333301571709038440102, −8.239660064429205271636037506011, −7.74948559321050562206883664486, −6.59102589022079319434487345337, −5.33384599195831313613862405588, −4.67026538534447522162326701852, −3.03338278247520545923998081395, −1.99578932799500149440106857892, −1.24590121522985050688774961464, 2.42477850293564150817510546859, 3.31076859360656331025693157071, 4.60748683617744662129392024423, 5.03529554286473481275362300825, 6.14514323347533231289354626282, 7.54174515636898227709327166624, 8.201469799784092583203539690609, 8.826743822826806275444047619249, 10.05784215803563567049979320905, 10.51853255787013952649841792780

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