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Properties

Label 2-722-19.11-c1-0-12
Degree $2$
Conductor $722$
Sign $0.853 - 0.520i$
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 + (0.939 − 1.62i)3-s + (−0.499 − 0.866i)4-s + (−1 + 1.73i)5-s + (0.939 + 1.62i)6-s + 5.06·7-s + 0.999·8-s + (−0.266 − 0.460i)9-s + (−0.999 − 1.73i)10-s − 1.41·11-s − 1.87·12-s + (0.652 + 1.13i)13-s + (−2.53 + 4.38i)14-s + (1.87 + 3.25i)15-s + (−0.5 + 0.866i)16-s + (−1.19 + 2.06i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + (0.542 − 0.939i)3-s + (−0.249 − 0.433i)4-s + (−0.447 + 0.774i)5-s + (0.383 + 0.664i)6-s + 1.91·7-s + 0.353·8-s + (−0.0886 − 0.153i)9-s + (−0.316 − 0.547i)10-s − 0.425·11-s − 0.542·12-s + (0.181 + 0.313i)13-s + (−0.676 + 1.17i)14-s + (0.485 + 0.840i)15-s + (−0.125 + 0.216i)16-s + (−0.289 + 0.501i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.853 - 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.853 - 0.520i$
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: \((2,\ 722,\ (\ :1/2),\ 0.853 - 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61554 + 0.453574i\)
\(L(\frac12)\) \(\approx\) \(1.61554 + 0.453574i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.939 + 1.62i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1 - 1.73i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 5.06T + 7T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + (-0.652 - 1.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (1.19 - 2.06i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (-1.53 - 2.65i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.22 + 7.32i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.369T + 31T^{2} \)
37 \( 1 - 4.82T + 37T^{2} \)
41 \( 1 + (-0.766 + 1.32i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-0.379 + 0.657i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-5.10 - 8.84i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (0.837 + 1.45i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.358 - 0.620i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.87 + 8.45i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.701 - 1.21i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-3.18 + 5.51i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-2.27 + 3.94i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-1.12 + 1.94i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 3.98T + 83T^{2} \)
89 \( 1 + (-5.32 - 9.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-0.766 + 1.32i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.76109916248802286984961268279, −9.329532332539206484595286269598, −8.328348449278171767433994944415, −7.74429092385787159232137995832, −7.42710490030343193929538406193, −6.33304573847833094426706499332, −5.16621122909524892779286257992, −4.12226092526031727896370259244, −2.43828171141272599897389947806, −1.44170510358132283446788503226, 1.13655825220601057003077281154, 2.57867777261666134303316123665, 3.93594139998928469755898451284, 4.65825449046446000792908430817, 5.27942919757935495160727419481, 7.26290881853055212840635263298, 8.213733365738639706806420550315, 8.639509799140265795158196204923, 9.344147283078552789374244799866, 10.49017078089219300075981081885

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