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Properties

Label 2-572-11.5-c1-0-10
Degree $2$
Conductor $572$
Sign $-0.269 + 0.962i$
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  + (0.178 + 0.129i)3-s + (1.35 − 4.15i)5-s + (−3.13 + 2.28i)7-s + (−0.912 − 2.80i)9-s + (3.23 + 0.746i)11-s + (−0.309 − 0.951i)13-s + (0.778 − 0.565i)15-s + (0.502 − 1.54i)17-s + (−3.94 − 2.86i)19-s − 0.854·21-s − 4.01·23-s + (−11.4 − 8.28i)25-s + (0.404 − 1.24i)27-s + (4.63 − 3.36i)29-s + (0.353 + 1.08i)31-s + ⋯
L(s)  = 1  + (0.102 + 0.0747i)3-s + (0.603 − 1.85i)5-s + (−1.18 + 0.862i)7-s + (−0.304 − 0.935i)9-s + (0.974 + 0.225i)11-s + (−0.0857 − 0.263i)13-s + (0.200 − 0.146i)15-s + (0.121 − 0.375i)17-s + (−0.904 − 0.656i)19-s − 0.186·21-s − 0.837·23-s + (−2.28 − 1.65i)25-s + (0.0779 − 0.239i)27-s + (0.861 − 0.625i)29-s + (0.0634 + 0.195i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 + 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.269 + 0.962i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (313, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.269 + 0.962i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.762497 - 1.00520i\)
\(L(\frac12)\) \(\approx\) \(0.762497 - 1.00520i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.23 - 0.746i)T \)
13 \( 1 + (0.309 + 0.951i)T \)
good3 \( 1 + (-0.178 - 0.129i)T + (0.927 + 2.85i)T^{2} \)
5 \( 1 + (-1.35 + 4.15i)T + (-4.04 - 2.93i)T^{2} \)
7 \( 1 + (3.13 - 2.28i)T + (2.16 - 6.65i)T^{2} \)
17 \( 1 + (-0.502 + 1.54i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (3.94 + 2.86i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 4.01T + 23T^{2} \)
29 \( 1 + (-4.63 + 3.36i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-0.353 - 1.08i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-2.50 + 1.82i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (-4.51 - 3.28i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 2.48T + 43T^{2} \)
47 \( 1 + (8.04 + 5.84i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.96 - 9.12i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-7.89 + 5.73i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (1.38 - 4.25i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 - 2.89T + 67T^{2} \)
71 \( 1 + (-3.96 + 12.2i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-3.66 + 2.66i)T + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (-2.14 - 6.59i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-0.949 + 2.92i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 16.3T + 89T^{2} \)
97 \( 1 + (-1.34 - 4.13i)T + (-78.4 + 57.0i)T^{2} \)
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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

−10.04600045626187501361317448039, −9.290955338931308052598739997614, −9.074852035246839111174610095997, −8.177976391184321307581903254605, −6.43400972494390657682673081028, −6.02303583168874649114213015191, −4.87883105950507078053498686482, −3.83721212175027157764583135439, −2.34937065232721064742639496052, −0.69367904943677613256173812352, 2.08204852794823742599186301089, 3.19501453880634180503319871044, 4.03809690697345177212907229855, 5.91901299779064426357385930784, 6.52310683066310607881960435888, 7.14451016862321924255779951584, 8.214822313187300256458549840308, 9.594062673294085673450353391655, 10.21067890854353582171102330268, 10.76046555932725334513324676422

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