↑

Properties

Label 2-2576-2576.1189-c0-0-1
Degree $2$
Conductor $2576$
Sign $-0.899 + 0.437i$
Analytic cond. $1.28559$
Root an. cond. $1.13383$
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  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (−0.281 − 0.959i)7-s + (−0.540 − 0.841i)8-s + (−0.909 − 0.415i)9-s + (−0.0303 + 0.424i)11-s + (−0.841 − 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.959 + 0.281i)18-s + (0.254 + 0.340i)22-s + (−0.415 − 0.909i)23-s + (−0.755 − 0.654i)25-s + (−0.989 + 0.142i)28-s + (−0.0855 − 0.114i)29-s + (−0.909 + 0.415i)32-s + ⋯
L(s)  = 1  + (0.755 − 0.654i)2-s + (0.142 − 0.989i)4-s + (−0.281 − 0.959i)7-s + (−0.540 − 0.841i)8-s + (−0.909 − 0.415i)9-s + (−0.0303 + 0.424i)11-s + (−0.841 − 0.540i)14-s + (−0.959 − 0.281i)16-s + (−0.959 + 0.281i)18-s + (0.254 + 0.340i)22-s + (−0.415 − 0.909i)23-s + (−0.755 − 0.654i)25-s + (−0.989 + 0.142i)28-s + (−0.0855 − 0.114i)29-s + (−0.909 + 0.415i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.899 + 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2576\)    =    \(2^{4} \cdot 7 \cdot 23\)
Sign: $-0.899 + 0.437i$
Analytic conductor: \(1.28559\)
Root analytic conductor: \(1.13383\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2576} (1189, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2576,\ (\ :0),\ -0.899 + 0.437i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.350036557\)
\(L(\frac12)\) \(\approx\) \(1.350036557\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.755 + 0.654i)T \)
7 \( 1 + (0.281 + 0.959i)T \)
23 \( 1 + (0.415 + 0.909i)T \)
good3 \( 1 + (0.909 + 0.415i)T^{2} \)
5 \( 1 + (0.755 + 0.654i)T^{2} \)
11 \( 1 + (0.0303 - 0.424i)T + (-0.989 - 0.142i)T^{2} \)
13 \( 1 + (-0.540 + 0.841i)T^{2} \)
17 \( 1 + (0.959 + 0.281i)T^{2} \)
19 \( 1 + (-0.281 - 0.959i)T^{2} \)
29 \( 1 + (0.0855 + 0.114i)T + (-0.281 + 0.959i)T^{2} \)
31 \( 1 + (-0.415 - 0.909i)T^{2} \)
37 \( 1 + (-1.64 + 0.613i)T + (0.755 - 0.654i)T^{2} \)
41 \( 1 + (-0.654 + 0.755i)T^{2} \)
43 \( 1 + (0.682 + 0.148i)T + (0.909 + 0.415i)T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 + (-1.75 - 0.956i)T + (0.540 + 0.841i)T^{2} \)
59 \( 1 + (0.540 - 0.841i)T^{2} \)
61 \( 1 + (-0.909 + 0.415i)T^{2} \)
67 \( 1 + (0.114 + 1.59i)T + (-0.989 + 0.142i)T^{2} \)
71 \( 1 + (-1.27 - 1.10i)T + (0.142 + 0.989i)T^{2} \)
73 \( 1 + (-0.959 + 0.281i)T^{2} \)
79 \( 1 + (1.45 + 0.425i)T + (0.841 + 0.540i)T^{2} \)
83 \( 1 + (0.755 - 0.654i)T^{2} \)
89 \( 1 + (0.415 - 0.909i)T^{2} \)
97 \( 1 + (0.654 - 0.755i)T^{2} \)
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

−8.914112506995795033471792469446, −7.960498747855395005415293048530, −7.01757062285388093199635792771, −6.26048437642480755641673839424, −5.64595718123053456772280907221, −4.50794685899921841038821005519, −3.98548322614164919109165469192, −3.02769608008742743391069378304, −2.13531350166818925985399340812, −0.64873839171118433975008474035, 2.14547747759485854689507570447, 2.98932227285499217755955207727, 3.77101820271249936652155947804, 4.93071127928668010979632636285, 5.68343661610601598141189475692, 6.00272392852467369801609708869, 7.00784539537830835221031332199, 7.928803207832490207065006888013, 8.429921951541453118742356265276, 9.157305563553326479930358097322

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