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Properties

Label 2-722-19.9-c1-0-12
Degree $2$
Conductor $722$
Sign $0.949 + 0.313i$
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 + (0.5 − 0.866i)7-s + (0.5 + 0.866i)8-s + (−0.347 + 1.96i)9-s + (3 + 5.19i)11-s + (−0.499 + 0.866i)12-s + (3.83 + 3.21i)13-s + (−0.939 − 0.342i)14-s + (0.766 − 0.642i)16-s + (0.520 + 2.95i)17-s + 2·18-s + (−0.173 − 0.984i)21-s + (4.59 − 3.85i)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 + (0.188 − 0.327i)7-s + (0.176 + 0.306i)8-s + (−0.115 + 0.656i)9-s + (0.904 + 1.56i)11-s + (−0.144 + 0.249i)12-s + (1.06 + 0.891i)13-s + (−0.251 − 0.0914i)14-s + (0.191 − 0.160i)16-s + (0.126 + 0.716i)17-s + 0.471·18-s + (−0.0378 − 0.214i)21-s + (0.979 − 0.822i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 + 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.949 + 0.313i$
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: \((2,\ 722,\ (\ :1/2),\ 0.949 + 0.313i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65161 - 0.265486i\)
\(L(\frac12)\) \(\approx\) \(1.65161 - 0.265486i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 + 0.866i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (-3 - 5.19i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.83 - 3.21i)T + (2.25 + 12.8i)T^{2} \)
17 \( 1 + (-0.520 - 2.95i)T + (-15.9 + 5.81i)T^{2} \)
23 \( 1 + (2.81 - 1.02i)T + (17.6 - 14.7i)T^{2} \)
29 \( 1 + (-1.56 + 8.86i)T + (-27.2 - 9.91i)T^{2} \)
31 \( 1 + (-2 + 3.46i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (7.11 - 40.3i)T^{2} \)
43 \( 1 + (7.51 + 2.73i)T + (32.9 + 27.6i)T^{2} \)
47 \( 1 + (-44.1 - 16.0i)T^{2} \)
53 \( 1 + (-2.81 + 1.02i)T + (40.6 - 34.0i)T^{2} \)
59 \( 1 + (-1.56 - 8.86i)T + (-55.4 + 20.1i)T^{2} \)
61 \( 1 + (-9.39 + 3.42i)T + (46.7 - 39.2i)T^{2} \)
67 \( 1 + (-0.868 + 4.92i)T + (-62.9 - 22.9i)T^{2} \)
71 \( 1 + (-5.63 - 2.05i)T + (54.3 + 45.6i)T^{2} \)
73 \( 1 + (5.36 - 4.49i)T + (12.6 - 71.8i)T^{2} \)
79 \( 1 + (7.66 - 6.42i)T + (13.7 - 77.7i)T^{2} \)
83 \( 1 + (-3 + 5.19i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (9.19 + 7.71i)T + (15.4 + 87.6i)T^{2} \)
97 \( 1 + (1.73 + 9.84i)T + (-91.1 + 33.1i)T^{2} \)
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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

−10.21067785969645760652692510027, −9.693731900426290115912599349462, −8.618335687856143581723337852666, −7.958551488641049551479082112669, −7.01802220954580161291552755924, −5.97964615773022321065727361789, −4.41375252937378254555112539149, −3.96171804786003108649136739563, −2.27413665242578004865924568394, −1.55818474055833538056788055642, 0.994987878445128122932214103324, 3.19386627895935498053782102512, 3.77765938451211476147794712054, 5.26048106056144150817686825500, 6.04476167328159055900701370646, 6.81781867058067644682453051994, 8.207976309412887826182227420666, 8.614259664507290877585102324232, 9.296232797736166783810351387223, 10.27029296520396596097533441447

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