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Properties

Label 2-572-11.4-c1-0-10
Degree $2$
Conductor $572$
Sign $-0.290 + 0.956i$
Analytic cond. $4.56744$
Root an. cond. $2.13715$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.405 − 1.24i)3-s + (1.75 − 1.27i)5-s + (−1.12 − 3.45i)7-s + (1.03 + 0.750i)9-s + (−1.01 − 3.15i)11-s + (−0.809 − 0.587i)13-s + (−0.881 − 2.71i)15-s + (−6.38 + 4.63i)17-s + (−0.410 + 1.26i)19-s − 4.77·21-s + 8.86·23-s + (−0.0860 + 0.264i)25-s + (4.54 − 3.29i)27-s + (−0.449 − 1.38i)29-s + (−3.60 − 2.61i)31-s + ⋯
L(s)  = 1  + (0.234 − 0.720i)3-s + (0.786 − 0.571i)5-s + (−0.424 − 1.30i)7-s + (0.344 + 0.250i)9-s + (−0.304 − 0.952i)11-s + (−0.224 − 0.163i)13-s + (−0.227 − 0.700i)15-s + (−1.54 + 1.12i)17-s + (−0.0942 + 0.290i)19-s − 1.04·21-s + 1.84·23-s + (−0.0172 + 0.0529i)25-s + (0.874 − 0.635i)27-s + (−0.0833 − 0.256i)29-s + (−0.647 − 0.470i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.290 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.290 + 0.956i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (521, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.290 + 0.956i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.967890 - 1.30541i\)
\(L(\frac12)\) \(\approx\) \(0.967890 - 1.30541i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (1.01 + 3.15i)T \)
13 \( 1 + (0.809 + 0.587i)T \)
good3 \( 1 + (-0.405 + 1.24i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.75 + 1.27i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.12 + 3.45i)T + (-5.66 + 4.11i)T^{2} \)
17 \( 1 + (6.38 - 4.63i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (0.410 - 1.26i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 8.86T + 23T^{2} \)
29 \( 1 + (0.449 + 1.38i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (3.60 + 2.61i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.231 + 0.713i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-1.95 + 6.01i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.99T + 43T^{2} \)
47 \( 1 + (2.27 - 7.00i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-1.93 - 1.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.952 + 2.93i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-8.36 + 6.07i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 7.64T + 67T^{2} \)
71 \( 1 + (-8.43 + 6.12i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.608 + 1.87i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-12.0 - 8.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (4.40 - 3.19i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.17T + 89T^{2} \)
97 \( 1 + (-12.3 - 8.94i)T + (29.9 + 92.2i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.68062701932430165329343190928, −9.523108018142222509772246669330, −8.705645788334970405921552845334, −7.72166399424674375373143608619, −6.90481018912983680498569599185, −6.06223607696823374418562497799, −4.85260634192625131207296598675, −3.71091135022700061161418302096, −2.19154514115572490328425865019, −0.922400146258374924568630681844, 2.24443524228328979019249320978, 2.97542467320358353623845664414, 4.55335064552925331874529951658, 5.31146964207623565738869124361, 6.59793013003390236712888557406, 7.11789486307851731901787620263, 8.911685953311139369166233538085, 9.204563313092334859427254167342, 9.960224806300864842657750278893, 10.81502964283817628583941038443

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