Properties

Label 2-722-19.11-c1-0-18
Degree $2$
Conductor $722$
Sign $0.0977 + 0.995i$
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 + (−0.618 + 1.07i)3-s + (−0.499 − 0.866i)4-s + (1.80 − 3.13i)5-s + (−0.618 − 1.07i)6-s − 3.23·7-s + 0.999·8-s + (0.736 + 1.27i)9-s + (1.80 + 3.13i)10-s − 3.23·11-s + 1.23·12-s + (0.690 + 1.19i)13-s + (1.61 − 2.80i)14-s + (2.23 + 3.87i)15-s + (−0.5 + 0.866i)16-s + (1.69 − 2.92i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (−0.356 + 0.618i)3-s + (−0.249 − 0.433i)4-s + (0.809 − 1.40i)5-s + (−0.252 − 0.437i)6-s − 1.22·7-s + 0.353·8-s + (0.245 + 0.424i)9-s + (0.572 + 0.990i)10-s − 0.975·11-s + 0.356·12-s + (0.191 + 0.331i)13-s + (0.432 − 0.749i)14-s + (0.577 + 1.00i)15-s + (−0.125 + 0.216i)16-s + (0.410 − 0.710i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0977 + 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.0977 + 0.995i$
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: \((2,\ 722,\ (\ :1/2),\ 0.0977 + 0.995i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.400631 - 0.363214i\)
\(L(\frac12)\) \(\approx\) \(0.400631 - 0.363214i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.618 - 1.07i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-1.80 + 3.13i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 3.23T + 7T^{2} \)
11 \( 1 + 3.23T + 11T^{2} \)
13 \( 1 + (-0.690 - 1.19i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-1.69 + 2.92i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (2.61 + 4.53i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.54 + 7.87i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 1.23T + 31T^{2} \)
37 \( 1 + 8.38T + 37T^{2} \)
41 \( 1 + (-0.427 + 0.739i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4.61 + 7.99i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.23 + 3.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (3.04 + 5.27i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.236 - 0.408i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (0.690 + 1.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.85 - 10.1i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.47 + 2.54i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.80 + 4.86i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.38 - 2.39i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 0.472T + 83T^{2} \)
89 \( 1 + (-4.42 - 7.66i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-4.30 + 7.46i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.933570432035646608838987499572, −9.490444379502354941766597764967, −8.633568216483189219307523179999, −7.70333640024198121601707650692, −6.52001672390653700584986518904, −5.55914904802321593811956206939, −5.08452195503890843772970455460, −4.00816511215324719900816029028, −2.17520127326718198609792065861, −0.31536143006747476238841005996, 1.67619233084467661411277240123, 2.96079781071457665273803340155, 3.55890543934995366841479222479, 5.58070517628957070579738600504, 6.29308833631848726773915605076, 7.05177637006031362485622541432, 7.83755637747858902804984012719, 9.289997340220463779340105531174, 9.909820885014623402826960822662, 10.54751986839632341443698419267

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