Properties

Label 2-24e2-576.133-c1-0-60
Degree $2$
Conductor $576$
Sign $0.920 + 0.389i$
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  + (−0.845 + 1.13i)2-s + (1.55 − 0.757i)3-s + (−0.568 − 1.91i)4-s + (−0.405 − 1.19i)5-s + (−0.459 + 2.40i)6-s + (0.538 + 0.702i)7-s + (2.65 + 0.977i)8-s + (1.85 − 2.35i)9-s + (1.69 + 0.551i)10-s + (1.72 − 0.113i)11-s + (−2.33 − 2.55i)12-s + (−0.136 − 0.155i)13-s + (−1.25 + 0.0166i)14-s + (−1.53 − 1.55i)15-s + (−3.35 + 2.18i)16-s + (−1.23 + 1.23i)17-s + ⋯
L(s)  = 1  + (−0.598 + 0.801i)2-s + (0.899 − 0.437i)3-s + (−0.284 − 0.958i)4-s + (−0.181 − 0.534i)5-s + (−0.187 + 0.982i)6-s + (0.203 + 0.265i)7-s + (0.938 + 0.345i)8-s + (0.617 − 0.786i)9-s + (0.537 + 0.174i)10-s + (0.520 − 0.0341i)11-s + (−0.674 − 0.737i)12-s + (−0.0378 − 0.0431i)13-s + (−0.334 + 0.00443i)14-s + (−0.397 − 0.401i)15-s + (−0.838 + 0.545i)16-s + (−0.300 + 0.300i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 576 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.920 + 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(576\)    =    \(2^{6} \cdot 3^{2}\)
Sign: $0.920 + 0.389i$
Analytic conductor: \(4.59938\)
Root analytic conductor: \(2.14461\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{576} (133, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 576,\ (\ :1/2),\ 0.920 + 0.389i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.40342 - 0.284873i\)
\(L(\frac12)\) \(\approx\) \(1.40342 - 0.284873i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.845 - 1.13i)T \)
3 \( 1 + (-1.55 + 0.757i)T \)
good5 \( 1 + (0.405 + 1.19i)T + (-3.96 + 3.04i)T^{2} \)
7 \( 1 + (-0.538 - 0.702i)T + (-1.81 + 6.76i)T^{2} \)
11 \( 1 + (-1.72 + 0.113i)T + (10.9 - 1.43i)T^{2} \)
13 \( 1 + (0.136 + 0.155i)T + (-1.69 + 12.8i)T^{2} \)
17 \( 1 + (1.23 - 1.23i)T - 17iT^{2} \)
19 \( 1 + (1.64 + 8.27i)T + (-17.5 + 7.27i)T^{2} \)
23 \( 1 + (-0.848 - 0.650i)T + (5.95 + 22.2i)T^{2} \)
29 \( 1 + (-3.48 - 7.06i)T + (-17.6 + 23.0i)T^{2} \)
31 \( 1 + (-1.77 + 1.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-1.88 + 9.47i)T + (-34.1 - 14.1i)T^{2} \)
41 \( 1 + (5.14 + 3.94i)T + (10.6 + 39.6i)T^{2} \)
43 \( 1 + (-0.193 - 2.95i)T + (-42.6 + 5.61i)T^{2} \)
47 \( 1 + (-2.26 - 8.46i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-0.236 - 0.354i)T + (-20.2 + 48.9i)T^{2} \)
59 \( 1 + (0.603 + 1.77i)T + (-46.8 + 35.9i)T^{2} \)
61 \( 1 + (-1.29 + 0.636i)T + (37.1 - 48.3i)T^{2} \)
67 \( 1 + (0.128 - 1.95i)T + (-66.4 - 8.74i)T^{2} \)
71 \( 1 + (3.07 - 7.42i)T + (-50.2 - 50.2i)T^{2} \)
73 \( 1 + (-6.36 - 15.3i)T + (-51.6 + 51.6i)T^{2} \)
79 \( 1 + (10.1 - 2.71i)T + (68.4 - 39.5i)T^{2} \)
83 \( 1 + (-2.23 - 0.758i)T + (65.8 + 50.5i)T^{2} \)
89 \( 1 + (-8.28 - 3.42i)T + (62.9 + 62.9i)T^{2} \)
97 \( 1 + (-3.16 - 1.82i)T + (48.5 + 84.0i)T^{2} \)
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

−10.44391100057400895806201447394, −9.200447358526014988131083406766, −8.901690940947789411817252605041, −8.175853214475412044377118669775, −7.11377725294425060503037116533, −6.53808454250865900895033829740, −5.15620037766837337688347342515, −4.17655296872074926031652868882, −2.46493183256066971086162926822, −1.00539999034171092251034345024, 1.64748295850038932511980984019, 2.90645151494299361826135857901, 3.80200131682920065735657602754, 4.67708924403381084437553611197, 6.54703891133829676337874928078, 7.62690836213024755820608222286, 8.268735595497823637095967024406, 9.102704083680267246060908904602, 10.10372227287310600268393627034, 10.42786354093341878437512218372

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