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Properties

Label 2-722-19.9-c1-0-21
Degree $2$
Conductor $722$
Sign $-0.624 + 0.780i$
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  + (−0.173 − 0.984i)2-s + (0.766 − 0.642i)3-s + (−0.939 + 0.342i)4-s + (−0.766 − 0.642i)6-s + (2 − 3.46i)7-s + (0.5 + 0.866i)8-s + (−0.347 + 1.96i)9-s + (−1.5 − 2.59i)11-s + (−0.499 + 0.866i)12-s + (1.53 + 1.28i)13-s + (−3.75 − 1.36i)14-s + (0.766 − 0.642i)16-s + (−1.04 − 5.90i)17-s + 2·18-s + (−0.694 − 3.93i)21-s + (−2.29 + 1.92i)22-s + ⋯
L(s)  = 1  + (−0.122 − 0.696i)2-s + (0.442 − 0.371i)3-s + (−0.469 + 0.171i)4-s + (−0.312 − 0.262i)6-s + (0.755 − 1.30i)7-s + (0.176 + 0.306i)8-s + (−0.115 + 0.656i)9-s + (−0.452 − 0.783i)11-s + (−0.144 + 0.249i)12-s + (0.424 + 0.356i)13-s + (−1.00 − 0.365i)14-s + (0.191 − 0.160i)16-s + (−0.252 − 1.43i)17-s + 0.471·18-s + (−0.151 − 0.859i)21-s + (−0.489 + 0.411i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.624 + 0.780i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $-0.624 + 0.780i$
Analytic conductor: \(5.76519\)
Root analytic conductor: \(2.40108\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{722} (389, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ -0.624 + 0.780i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.670585 - 1.39563i\)
\(L(\frac12)\) \(\approx\) \(0.670585 - 1.39563i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.766 + 0.642i)T + (0.520 - 2.95i)T^{2} \)
5 \( 1 + (3.83 + 3.21i)T^{2} \)
7 \( 1 + (-2 + 3.46i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (1.5 + 2.59i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-1.53 - 1.28i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (1.04 + 5.90i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (-5.63 + 2.05i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (1 - 1.73i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 10T + 37T^{2} \)
41 \( 1 + (-6.89 + 5.78i)T + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (-3.75 - 1.36i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (5.63 - 2.05i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (1.56 + 8.86i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-3.75 + 1.36i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (1.21 - 6.89i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.63 - 2.05i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (0.766 - 0.642i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (3.06 - 2.57i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (1.5 - 2.59i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-4.59 - 3.85i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (-2.95 - 16.7i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.37816980404250858939049276462, −9.182621967412363537096081683243, −8.394384413397975443100235133203, −7.63445468891621906521852493605, −6.91270962841848666749073843275, −5.30188204568022075377504544624, −4.47276060416477687712057797388, −3.31272950090512979660215625329, −2.18196849201305948438451788655, −0.836217416795624948279877900830, 1.82552688225243442739535305241, 3.25649857975766699044802385393, 4.44507946815948658660868714569, 5.46988346462318930481884340639, 6.14898461152331456675878561567, 7.38088595300721525070545633461, 8.290532561926684936637317863431, 8.872230304508189551622566938287, 9.540973754406416879272643321373, 10.56070108354494410392035512294

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