Properties

Label 2-722-19.11-c1-0-20
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 + (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

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

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

\(L(1)\) \(\approx\) \(1.37713 - 1.24851i\)
\(L(\frac12)\) \(\approx\) \(1.37713 - 1.24851i\)
\(L(\frac{3}{2})\) 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

−9.722768884471553237262789586814, −8.892652286406122515921258923660, −8.557947737408889231232239853999, −7.48718246076117438796055171904, −7.00164771604062078997721534675, −6.08800207767426882333125787534, −4.99643510671926578728302809580, −3.40525539001598431616560734667, −1.93638475503339328773278316427, −1.08353751601116611642036280093, 1.98269953821665039382745306698, 3.22031965290119469632352592551, 3.77048739167155032905645609999, 4.87640257880231938157291590842, 5.91792496657641982315668487452, 7.56770578914605770805297710412, 8.487809243160306838944311675233, 8.887081002097575274084208641853, 10.07989407992752115526182645216, 10.34057010073549668600257539055

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