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Properties

Label 2-722-19.11-c1-0-1
Degree $2$
Conductor $722$
Sign $-0.813 + 0.582i$
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 + (−1.32 + 2.29i)3-s + (−0.499 − 0.866i)4-s + (−1.82 + 3.15i)5-s + (1.32 + 2.29i)6-s − 1.64·7-s − 0.999·8-s + (−2 − 3.46i)9-s + (1.82 + 3.15i)10-s + 0.645·11-s + 2.64·12-s + (1 + 1.73i)13-s + (−0.822 + 1.42i)14-s + (−4.82 − 8.35i)15-s + (−0.5 + 0.866i)16-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s + (−0.763 + 1.32i)3-s + (−0.249 − 0.433i)4-s + (−0.815 + 1.41i)5-s + (0.540 + 0.935i)6-s − 0.622·7-s − 0.353·8-s + (−0.666 − 1.15i)9-s + (0.576 + 0.998i)10-s + 0.194·11-s + 0.763·12-s + (0.277 + 0.480i)13-s + (−0.219 + 0.380i)14-s + (−1.24 − 2.15i)15-s + (−0.125 + 0.216i)16-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.813 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.813 + 0.582i$
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: \((2,\ 722,\ (\ :1/2),\ -0.813 + 0.582i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0637450 - 0.198490i\)
\(L(\frac12)\) \(\approx\) \(0.0637450 - 0.198490i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.32 - 2.29i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 + (1.82 - 3.15i)T + (-2.5 - 4.33i)T^{2} \)
7 \( 1 + 1.64T + 7T^{2} \)
11 \( 1 - 0.645T + 11T^{2} \)
13 \( 1 + (-1 - 1.73i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
23 \( 1 + (1.82 + 3.15i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.82 + 3.15i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 0.354T + 31T^{2} \)
37 \( 1 + 5.64T + 37T^{2} \)
41 \( 1 + (-5.14 + 8.91i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.354 - 0.613i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.82 + 8.35i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.29 - 7.43i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-3.96 + 6.87i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-7.46 - 12.9i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.32 + 4.02i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6.64 - 11.5i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (6.14 - 10.6i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2 - 3.46i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 7.93T + 83T^{2} \)
89 \( 1 + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (7.14 - 12.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.84293712861694123206839292985, −10.32290128233215160771869518867, −9.692460792757669479508922379034, −8.622501450042383234431722743789, −7.16054967659228655629257459698, −6.36178782700391729296354107165, −5.43009465454908604904062406006, −4.07962529356258584985091224826, −3.79469002349440965895429308802, −2.63767600579043564507635117181, 0.11112275679238661711742271171, 1.39837282818791753950657703727, 3.43233674483171500953624746537, 4.64841488790472037998319416795, 5.55657511168297616785147827481, 6.30496928475211626905065113404, 7.25851113022232022246912887930, 7.942545757317201351345527902149, 8.668283161446809619657511118256, 9.681691621635801803494374290324

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