↑

Properties

Label 2-3520-3520.3189-c0-0-0
Degree $2$
Conductor $3520$
Sign $0.471 - 0.881i$
Analytic cond. $1.75670$
Root an. cond. $1.32540$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.956 − 0.290i)2-s + (0.831 + 0.555i)4-s + (0.555 + 0.831i)5-s + (1.17 + 0.485i)7-s + (−0.634 − 0.773i)8-s + (0.923 − 0.382i)9-s + (−0.290 − 0.956i)10-s + (−0.195 + 0.980i)11-s + (−0.523 + 0.783i)13-s + (−0.980 − 0.804i)14-s + (0.382 + 0.923i)16-s + (−0.138 + 0.138i)17-s + (−0.995 + 0.0980i)18-s + i·20-s + (0.471 − 0.881i)22-s + ⋯
L(s)  = 1  + (−0.956 − 0.290i)2-s + (0.831 + 0.555i)4-s + (0.555 + 0.831i)5-s + (1.17 + 0.485i)7-s + (−0.634 − 0.773i)8-s + (0.923 − 0.382i)9-s + (−0.290 − 0.956i)10-s + (−0.195 + 0.980i)11-s + (−0.523 + 0.783i)13-s + (−0.980 − 0.804i)14-s + (0.382 + 0.923i)16-s + (−0.138 + 0.138i)17-s + (−0.995 + 0.0980i)18-s + i·20-s + (0.471 − 0.881i)22-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3520 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3520\)    =    \(2^{6} \cdot 5 \cdot 11\)
Sign: $0.471 - 0.881i$
Analytic conductor: \(1.75670\)
Root analytic conductor: \(1.32540\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3520} (3189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3520,\ (\ :0),\ 0.471 - 0.881i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.077012772\)
\(L(\frac12)\) \(\approx\) \(1.077012772\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.956 + 0.290i)T \)
5 \( 1 + (-0.555 - 0.831i)T \)
11 \( 1 + (0.195 - 0.980i)T \)
good3 \( 1 + (-0.923 + 0.382i)T^{2} \)
7 \( 1 + (-1.17 - 0.485i)T + (0.707 + 0.707i)T^{2} \)
13 \( 1 + (0.523 - 0.783i)T + (-0.382 - 0.923i)T^{2} \)
17 \( 1 + (0.138 - 0.138i)T - iT^{2} \)
19 \( 1 + (-0.382 - 0.923i)T^{2} \)
23 \( 1 + (0.707 - 0.707i)T^{2} \)
29 \( 1 + (0.923 - 0.382i)T^{2} \)
31 \( 1 - 0.390iT - T^{2} \)
37 \( 1 + (-0.382 + 0.923i)T^{2} \)
41 \( 1 + (0.707 - 0.707i)T^{2} \)
43 \( 1 + (0.569 + 0.113i)T + (0.923 + 0.382i)T^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (0.923 + 0.382i)T^{2} \)
59 \( 1 + (0.617 + 0.923i)T + (-0.382 + 0.923i)T^{2} \)
61 \( 1 + (-0.923 + 0.382i)T^{2} \)
67 \( 1 + (-0.923 + 0.382i)T^{2} \)
71 \( 1 + (1.70 + 0.707i)T + (0.707 + 0.707i)T^{2} \)
73 \( 1 + (-1.76 + 0.732i)T + (0.707 - 0.707i)T^{2} \)
79 \( 1 - iT^{2} \)
83 \( 1 + (0.783 + 0.523i)T + (0.382 + 0.923i)T^{2} \)
89 \( 1 + (-0.636 + 1.53i)T + (-0.707 - 0.707i)T^{2} \)
97 \( 1 + T^{2} \)
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

−9.072477294276752172450661923005, −8.121690500658041618219227762071, −7.39390134864752216500633625743, −6.91758686051124662222598455051, −6.19718242388769471740482168660, −5.04187861193804525886499971819, −4.23305509148495909740681829959, −3.07498835412212048365522659091, −1.94409736924169469456526110623, −1.72891024862184311166258493607, 0.902667298673118719239781102385, 1.69006320796559485785751516705, 2.69672706220845820531463175835, 4.18621667354165464294805468683, 5.08648339914291181305058290522, 5.53323336827737981580714892475, 6.52067446979536565588652126596, 7.45026793511275067876836973894, 7.995135893089921270822704708740, 8.463854859900055496805854720920

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