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Properties

Label 2-722-19.11-c1-0-14
Degree $2$
Conductor $722$
Sign $0.705 + 0.708i$
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 + (−1.61 + 2.80i)3-s + (−0.499 − 0.866i)4-s + (0.690 − 1.19i)5-s + (1.61 + 2.80i)6-s + 1.23·7-s − 0.999·8-s + (−3.73 − 6.47i)9-s + (−0.690 − 1.19i)10-s + 1.23·11-s + 3.23·12-s + (−1.80 − 3.13i)13-s + (0.618 − 1.07i)14-s + (2.23 + 3.87i)15-s + (−0.5 + 0.866i)16-s + (2.80 − 4.86i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.934 + 1.61i)3-s + (−0.249 − 0.433i)4-s + (0.309 − 0.535i)5-s + (0.660 + 1.14i)6-s + 0.467·7-s − 0.353·8-s + (−1.24 − 2.15i)9-s + (−0.218 − 0.378i)10-s + 0.372·11-s + 0.934·12-s + (−0.501 − 0.869i)13-s + (0.165 − 0.286i)14-s + (0.577 + 1.00i)15-s + (−0.125 + 0.216i)16-s + (0.681 − 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.705 + 0.708i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.705 + 0.708i$
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: \((2,\ 722,\ (\ :1/2),\ 0.705 + 0.708i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.19978 - 0.498391i\)
\(L(\frac12)\) \(\approx\) \(1.19978 - 0.498391i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.61 - 2.80i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (-0.690 + 1.19i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 - 1.23T + 7T^{2} \)
11 \( 1 - 1.23T + 11T^{2} \)
13 \( 1 + (1.80 + 3.13i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-2.80 + 4.86i)T + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (0.381 + 0.661i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.04 + 1.81i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 3.23T + 31T^{2} \)
37 \( 1 - 10.6T + 37T^{2} \)
41 \( 1 + (-2.92 + 5.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.38 + 4.12i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.23 - 3.87i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.54 + 4.40i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.23 - 7.33i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (1.80 + 3.13i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.854 - 1.47i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-7.47 + 12.9i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-1.69 + 2.92i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-3.61 + 6.26i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 8.47T + 83T^{2} \)
89 \( 1 + (1.07 + 1.85i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (3.19 - 5.52i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.44631985503955603274417057109, −9.443247428866416990457404284274, −9.310418209403036185536946296029, −7.85367605392379471681955191495, −6.25275186184786870581959048017, −5.31744284664215707895612078276, −4.92581144253103095225912929973, −3.99769621503251221926849775689, −2.87321467155649430449686514562, −0.76215446301263308133281992788, 1.37181348857779144558665479826, 2.57862314656426544821554652417, 4.37983955481173885225284467108, 5.49900380083006096075360209011, 6.32647723257722720433345763864, 6.73265433128176795421406448344, 7.73090460905114395242527030502, 8.246163003534852252910305008740, 9.644196077393863931832452680316, 10.88271906160889347084941030233

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