↑

Properties

Label 2-3584-3584.1525-c0-0-0
Degree $2$
Conductor $3584$
Sign $0.960 + 0.278i$
Analytic cond. $1.78864$
Root an. cond. $1.33740$
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  + (0.595 − 0.803i)2-s + (−0.290 − 0.956i)4-s + (0.903 + 0.427i)7-s + (−0.941 − 0.336i)8-s + (0.0490 + 0.998i)9-s + (0.223 + 1.28i)11-s + (0.881 − 0.471i)14-s + (−0.831 + 0.555i)16-s + (0.831 + 0.555i)18-s + (1.16 + 0.587i)22-s + (0.882 + 1.19i)23-s + (−0.514 − 0.857i)25-s + (0.146 − 0.989i)28-s + (−0.285 + 0.450i)29-s + (−0.0490 + 0.998i)32-s + ⋯
L(s)  = 1  + (0.595 − 0.803i)2-s + (−0.290 − 0.956i)4-s + (0.903 + 0.427i)7-s + (−0.941 − 0.336i)8-s + (0.0490 + 0.998i)9-s + (0.223 + 1.28i)11-s + (0.881 − 0.471i)14-s + (−0.831 + 0.555i)16-s + (0.831 + 0.555i)18-s + (1.16 + 0.587i)22-s + (0.882 + 1.19i)23-s + (−0.514 − 0.857i)25-s + (0.146 − 0.989i)28-s + (−0.285 + 0.450i)29-s + (−0.0490 + 0.998i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3584 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 + 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3584\)    =    \(2^{9} \cdot 7\)
Sign: $0.960 + 0.278i$
Analytic conductor: \(1.78864\)
Root analytic conductor: \(1.33740\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3584} (1525, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3584,\ (\ :0),\ 0.960 + 0.278i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.806067924\)
\(L(\frac12)\) \(\approx\) \(1.806067924\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.595 + 0.803i)T \)
7 \( 1 + (-0.903 - 0.427i)T \)
good3 \( 1 + (-0.0490 - 0.998i)T^{2} \)
5 \( 1 + (0.514 + 0.857i)T^{2} \)
11 \( 1 + (-0.223 - 1.28i)T + (-0.941 + 0.336i)T^{2} \)
13 \( 1 + (0.242 + 0.970i)T^{2} \)
17 \( 1 + (0.831 + 0.555i)T^{2} \)
19 \( 1 + (0.146 - 0.989i)T^{2} \)
23 \( 1 + (-0.882 - 1.19i)T + (-0.290 + 0.956i)T^{2} \)
29 \( 1 + (0.285 - 0.450i)T + (-0.427 - 0.903i)T^{2} \)
31 \( 1 + (0.382 - 0.923i)T^{2} \)
37 \( 1 + (0.107 - 0.324i)T + (-0.803 - 0.595i)T^{2} \)
41 \( 1 + (0.471 + 0.881i)T^{2} \)
43 \( 1 + (0.0107 - 0.438i)T + (-0.998 - 0.0490i)T^{2} \)
47 \( 1 + (-0.195 - 0.980i)T^{2} \)
53 \( 1 + (-0.433 - 0.684i)T + (-0.427 + 0.903i)T^{2} \)
59 \( 1 + (-0.242 + 0.970i)T^{2} \)
61 \( 1 + (-0.740 - 0.671i)T^{2} \)
67 \( 1 + (0.810 + 1.82i)T + (-0.671 + 0.740i)T^{2} \)
71 \( 1 + (-1.47 + 1.33i)T + (0.0980 - 0.995i)T^{2} \)
73 \( 1 + (-0.634 + 0.773i)T^{2} \)
79 \( 1 + (-0.190 + 1.93i)T + (-0.980 - 0.195i)T^{2} \)
83 \( 1 + (0.803 - 0.595i)T^{2} \)
89 \( 1 + (0.290 + 0.956i)T^{2} \)
97 \( 1 + (-0.923 - 0.382i)T^{2} \)
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

−8.894851337204484803096396272004, −7.910640375610055447622534135169, −7.31098644743271837639334042711, −6.27810554609250779777239080082, −5.33835872150682724896507668489, −4.83989278660838686058667782921, −4.24149646959203050069860407405, −3.08434376478718349298594368209, −2.07177890909320600870898448826, −1.57130756721084093371427476978, 0.930736360697873645677866377349, 2.57225878907819463467887304650, 3.63959596783203079253013858707, 4.07692656887393409196381530005, 5.14065875443496499558678632222, 5.73132523924519872650736482982, 6.57706072440693166967863714342, 7.13089383610648307843979362397, 8.004208792868816871102580660411, 8.638197128153193745798734430197

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