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Properties

Label 2-722-19.7-c1-0-12
Degree $2$
Conductor $722$
Sign $0.321 - 0.946i$
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 + (−0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + (2 + 3.46i)5-s + (0.499 − 0.866i)6-s + 3·7-s − 0.999·8-s + (1 − 1.73i)9-s + (−1.99 + 3.46i)10-s + 2·11-s + 0.999·12-s + (−0.5 + 0.866i)13-s + (1.5 + 2.59i)14-s + (1.99 − 3.46i)15-s + (−0.5 − 0.866i)16-s + (−1.5 − 2.59i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (−0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + (0.894 + 1.54i)5-s + (0.204 − 0.353i)6-s + 1.13·7-s − 0.353·8-s + (0.333 − 0.577i)9-s + (−0.632 + 1.09i)10-s + 0.603·11-s + 0.288·12-s + (−0.138 + 0.240i)13-s + (0.400 + 0.694i)14-s + (0.516 − 0.894i)15-s + (−0.125 − 0.216i)16-s + (−0.363 − 0.630i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.321 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.321 - 0.946i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (653, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.321 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.75771 + 1.25880i\)
\(L(\frac12)\) \(\approx\) \(1.75771 + 1.25880i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T + (-1.5 + 2.59i)T^{2} \)
5 \( 1 + (-2 - 3.46i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 - 3T + 7T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (0.5 - 0.866i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + (1.5 + 2.59i)T + (-8.5 + 14.7i)T^{2} \)
23 \( 1 + (-0.5 + 0.866i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.5 - 4.33i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (4 + 6.92i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (2 + 3.46i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (4 - 6.92i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-7.5 - 12.9i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (1 - 1.73i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.5 + 2.59i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (-1 - 1.73i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.5 + 7.79i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (5 + 8.66i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6T + 83T^{2} \)
89 \( 1 + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (1 + 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.65387744585778914815940475328, −9.721197812965410889530907094255, −8.808739002627495039359947268944, −7.54755396555116167771058538259, −6.91125496287047592672228205417, −6.36589902896581228285441940113, −5.43941584867101499350562865167, −4.24562218562692619176433309802, −2.93813734857464144984945632669, −1.67797095909919273319346332353, 1.26233499473525864654899520538, 2.10106941062416035677128623761, 4.08908822754357497010806173823, 4.80521822444863616712041183249, 5.28574193033956699311841218684, 6.31576277241842489269853877344, 8.044718166585263193492047048211, 8.584044633026906667800329644785, 9.712076008961357665277586307698, 10.05929815062021394255884893031

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