↑

Properties

Label 2-24e2-576.11-c1-0-75
Degree $2$
Conductor $576$
Sign $-0.954 + 0.296i$
Analytic cond. $4.59938$
Root an. cond. $2.14461$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−1.24 − 0.675i)2-s + (1.24 − 1.20i)3-s + (1.08 + 1.67i)4-s + (−0.558 − 0.275i)5-s + (−2.36 + 0.650i)6-s + (−2.33 − 3.04i)7-s + (−0.216 − 2.82i)8-s + (0.111 − 2.99i)9-s + (0.508 + 0.719i)10-s + (0.0326 + 0.0286i)11-s + (3.37 + 0.787i)12-s + (6.07 + 0.398i)13-s + (0.845 + 5.36i)14-s + (−1.02 + 0.327i)15-s + (−1.63 + 3.64i)16-s + (−4.66 − 4.66i)17-s + ⋯
L(s)  = 1  + (−0.878 − 0.477i)2-s + (0.720 − 0.693i)3-s + (0.543 + 0.839i)4-s + (−0.249 − 0.123i)5-s + (−0.964 + 0.265i)6-s + (−0.883 − 1.15i)7-s + (−0.0764 − 0.997i)8-s + (0.0370 − 0.999i)9-s + (0.160 + 0.227i)10-s + (0.00984 + 0.00863i)11-s + (0.973 + 0.227i)12-s + (1.68 + 0.110i)13-s + (0.225 + 1.43i)14-s + (−0.265 + 0.0846i)15-s + (−0.409 + 0.912i)16-s + (−1.13 − 1.13i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.954 + 0.296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $-0.954 + 0.296i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ -0.954 + 0.296i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128405 - 0.845716i\)
\(L(\frac12)\) \(\approx\) \(0.128405 - 0.845716i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.675i)T \)
3 \( 1 + (-1.24 + 1.20i)T \)
good5 \( 1 + (0.558 + 0.275i)T + (3.04 + 3.96i)T^{2} \)
7 \( 1 + (2.33 + 3.04i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (-0.0326 - 0.0286i)T + (1.43 + 10.9i)T^{2} \)
13 \( 1 + (-6.07 - 0.398i)T + (12.8 + 1.69i)T^{2} \)
17 \( 1 + (4.66 + 4.66i)T + 17iT^{2} \)
19 \( 1 + (3.07 - 4.60i)T + (-7.27 - 17.5i)T^{2} \)
23 \( 1 + (1.73 - 2.26i)T + (-5.95 - 22.2i)T^{2} \)
29 \( 1 + (0.252 - 0.745i)T + (-23.0 - 17.6i)T^{2} \)
31 \( 1 + (-2.56 - 4.44i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.90 + 4.61i)T + (14.1 - 34.1i)T^{2} \)
41 \( 1 + (8.87 + 6.80i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (2.62 + 2.30i)T + (5.61 + 42.6i)T^{2} \)
47 \( 1 + (1.26 + 4.73i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (6.63 + 1.31i)T + (48.9 + 20.2i)T^{2} \)
59 \( 1 + (-1.12 + 2.28i)T + (-35.9 - 46.8i)T^{2} \)
61 \( 1 + (-1.85 + 5.46i)T + (-48.3 - 37.1i)T^{2} \)
67 \( 1 + (0.0976 - 0.0856i)T + (8.74 - 66.4i)T^{2} \)
71 \( 1 + (-6.63 - 2.74i)T + (50.2 + 50.2i)T^{2} \)
73 \( 1 + (-7.99 + 3.31i)T + (51.6 - 51.6i)T^{2} \)
79 \( 1 + (-3.43 - 12.8i)T + (-68.4 + 39.5i)T^{2} \)
83 \( 1 + (-1.34 + 0.662i)T + (50.5 - 65.8i)T^{2} \)
89 \( 1 + (-12.4 - 5.15i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (1.47 + 0.853i)T + (48.5 + 84.0i)T^{2} \)
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

−10.22592396922797904132508741091, −9.389382939046902428995332629040, −8.536506687648533795100974979147, −7.889842926475451357881072078925, −6.82739062568759876547780341608, −6.39757812853148338435879202172, −3.96381167814083420875468241392, −3.45956698565770974756948024870, −1.97292213571518100572959122693, −0.57296445816663763511414647159, 2.08395525654999974228349721840, 3.22940122905609103489751104381, 4.56021400134632969704229472218, 6.05233506930299725955337093488, 6.45321811928209570303296673985, 8.020240774723928547870990882789, 8.591344206409027520692086699787, 9.167944746486891219286422315244, 9.983909046303900789896443076278, 10.95017961390779963832116184810

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