Properties

Label 2-722-1.1-c1-0-21
Degree $2$
Conductor $722$
Sign $-1$
Analytic cond. $5.76519$
Root an. cond. $2.40108$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  − 2-s − 1.28·3-s + 4-s + 3.69·5-s + 1.28·6-s − 0.442·7-s − 8-s − 1.35·9-s − 3.69·10-s − 4.02·11-s − 1.28·12-s − 4.89·13-s + 0.442·14-s − 4.74·15-s + 16-s + 0.266·17-s + 1.35·18-s + 3.69·20-s + 0.568·21-s + 4.02·22-s − 9.20·23-s + 1.28·24-s + 8.65·25-s + 4.89·26-s + 5.58·27-s − 0.442·28-s + 0.223·29-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.741·3-s + 0.5·4-s + 1.65·5-s + 0.524·6-s − 0.167·7-s − 0.353·8-s − 0.450·9-s − 1.16·10-s − 1.21·11-s − 0.370·12-s − 1.35·13-s + 0.118·14-s − 1.22·15-s + 0.250·16-s + 0.0647·17-s + 0.318·18-s + 0.826·20-s + 0.123·21-s + 0.859·22-s − 1.92·23-s + 0.262·24-s + 1.73·25-s + 0.959·26-s + 1.07·27-s − 0.0836·28-s + 0.0415·29-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
19 \( 1 \)
good3 \( 1 + 1.28T + 3T^{2} \)
5 \( 1 - 3.69T + 5T^{2} \)
7 \( 1 + 0.442T + 7T^{2} \)
11 \( 1 + 4.02T + 11T^{2} \)
13 \( 1 + 4.89T + 13T^{2} \)
17 \( 1 - 0.266T + 17T^{2} \)
23 \( 1 + 9.20T + 23T^{2} \)
29 \( 1 - 0.223T + 29T^{2} \)
31 \( 1 + 3.47T + 31T^{2} \)
37 \( 1 + 1.44T + 37T^{2} \)
41 \( 1 - 7.86T + 41T^{2} \)
43 \( 1 + 5.63T + 43T^{2} \)
47 \( 1 - 2.19T + 47T^{2} \)
53 \( 1 + 9.94T + 53T^{2} \)
59 \( 1 - 3.50T + 59T^{2} \)
61 \( 1 - 4.07T + 61T^{2} \)
67 \( 1 + 0.147T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 1.42T + 73T^{2} \)
79 \( 1 + 10.2T + 79T^{2} \)
83 \( 1 + 3.28T + 83T^{2} \)
89 \( 1 - 5.96T + 89T^{2} \)
97 \( 1 + 4.08T + 97T^{2} \)
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

−10.04121409479381863619536832668, −9.420917843180723380294516036132, −8.318104732015153053729933299732, −7.35338294108604237203115599098, −6.24146702595650531911603152424, −5.68885513654443510405265218553, −4.91144723313879746597015338512, −2.79313639399542843116978209097, −1.96328859067998024308222298758, 0, 1.96328859067998024308222298758, 2.79313639399542843116978209097, 4.91144723313879746597015338512, 5.68885513654443510405265218553, 6.24146702595650531911603152424, 7.35338294108604237203115599098, 8.318104732015153053729933299732, 9.420917843180723380294516036132, 10.04121409479381863619536832668

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