Properties

Degree $2$
Conductor $722$
Sign $0.999 + 0.00130i$
Motivic weight $1$
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more about

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (3.75 − 1.36i)5-s + (−0.766 + 0.642i)6-s + (−1.5 − 2.59i)7-s + (0.5 − 0.866i)8-s + (−0.347 − 1.96i)9-s + (0.694 + 3.93i)10-s + (−1 + 1.73i)11-s + (−0.499 − 0.866i)12-s + (0.766 − 0.642i)13-s + (2.81 − 1.02i)14-s + (3.75 + 1.36i)15-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s + ⋯
L(s)  = 1  + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (1.68 − 0.611i)5-s + (−0.312 + 0.262i)6-s + (−0.566 − 0.981i)7-s + (0.176 − 0.306i)8-s + (−0.115 − 0.656i)9-s + (0.219 + 1.24i)10-s + (−0.301 + 0.522i)11-s + (−0.144 − 0.249i)12-s + (0.212 − 0.178i)13-s + (0.753 − 0.274i)14-s + (0.970 + 0.353i)15-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.00130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(722\)    =    \(2 \cdot 19^{2}\)
Sign: $0.999 + 0.00130i$
Motivic weight: \(1\)
Character: $\chi_{722} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 722,\ (\ :1/2),\ 0.999 + 0.00130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.90907 - 0.00124606i\)
\(L(\frac12)\) \(\approx\) \(1.90907 - 0.00124606i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.75 + 1.36i)T + (3.83 - 3.21i)T^{2} \)
7 \( 1 + (1.5 + 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-0.766 + 0.642i)T + (2.25 - 12.8i)T^{2} \)
17 \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \)
23 \( 1 + (-0.939 - 0.342i)T + (17.6 + 14.7i)T^{2} \)
29 \( 1 + (-0.868 - 4.92i)T + (-27.2 + 9.91i)T^{2} \)
31 \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2T + 37T^{2} \)
41 \( 1 + (-6.12 - 5.14i)T + (7.11 + 40.3i)T^{2} \)
43 \( 1 + (3.75 - 1.36i)T + (32.9 - 27.6i)T^{2} \)
47 \( 1 + (-1.38 - 7.87i)T + (-44.1 + 16.0i)T^{2} \)
53 \( 1 + (0.939 + 0.342i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (2.60 - 14.7i)T + (-55.4 - 20.1i)T^{2} \)
61 \( 1 + (1.87 + 0.684i)T + (46.7 + 39.2i)T^{2} \)
67 \( 1 + (0.520 + 2.95i)T + (-62.9 + 22.9i)T^{2} \)
71 \( 1 + (-1.87 + 0.684i)T + (54.3 - 45.6i)T^{2} \)
73 \( 1 + (-6.89 - 5.78i)T + (12.6 + 71.8i)T^{2} \)
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 \( 1 + (15.4 - 87.6i)T^{2} \)
97 \( 1 + (-0.347 + 1.96i)T + (-91.1 - 33.1i)T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.946385458866876128409090674017, −9.505712989964111900621959079436, −8.993701741030379317684023459054, −7.79434317503600615902412098691, −6.77353212831371083229918617664, −6.03628928960551209776478725869, −5.11038837896771099169192459280, −4.09033629433171989046159045923, −2.73782565787091805092682040866, −1.05577232352212990458060962920, 1.80585112243372026663086147301, 2.47006291817472578480119666090, 3.33047867291009075337380405714, 5.20177896065939709604276542391, 5.88854829165250777241906922117, 6.75815530400860985243215398441, 8.086342359142549424141074073998, 8.942395444038279752766556746361, 9.523242667902387828733021450479, 10.49156269761228305893753983231

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