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Properties

Label 2-572-572.283-c1-0-22
Degree $2$
Conductor $572$
Sign $-0.0751 - 0.997i$
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  + (−1.10 + 0.886i)2-s + (0.133 + 0.120i)3-s + (0.427 − 1.95i)4-s + (−1.12 + 1.55i)5-s + (−0.253 − 0.0140i)6-s + (−0.344 + 0.310i)7-s + (1.26 + 2.53i)8-s + (−0.310 − 2.95i)9-s + (−0.133 − 2.70i)10-s + (3.24 − 0.704i)11-s + (0.291 − 0.209i)12-s + (1.86 + 3.08i)13-s + (0.104 − 0.647i)14-s + (−0.336 + 0.0716i)15-s + (−3.63 − 1.67i)16-s + (1.89 + 4.26i)17-s + ⋯
L(s)  = 1  + (−0.779 + 0.626i)2-s + (0.0770 + 0.0693i)3-s + (0.213 − 0.976i)4-s + (−0.504 + 0.694i)5-s + (−0.103 − 0.00574i)6-s + (−0.130 + 0.117i)7-s + (0.445 + 0.895i)8-s + (−0.103 − 0.983i)9-s + (−0.0422 − 0.856i)10-s + (0.977 − 0.212i)11-s + (0.0842 − 0.0604i)12-s + (0.517 + 0.855i)13-s + (0.0279 − 0.173i)14-s + (−0.0869 + 0.0184i)15-s + (−0.908 − 0.418i)16-s + (0.460 + 1.03i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0751 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0751 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.0751 - 0.997i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (283, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.0751 - 0.997i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.617816 + 0.666129i\)
\(L(\frac12)\) \(\approx\) \(0.617816 + 0.666129i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.10 - 0.886i)T \)
11 \( 1 + (-3.24 + 0.704i)T \)
13 \( 1 + (-1.86 - 3.08i)T \)
good3 \( 1 + (-0.133 - 0.120i)T + (0.313 + 2.98i)T^{2} \)
5 \( 1 + (1.12 - 1.55i)T + (-1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.344 - 0.310i)T + (0.731 - 6.96i)T^{2} \)
17 \( 1 + (-1.89 - 4.26i)T + (-11.3 + 12.6i)T^{2} \)
19 \( 1 + (-0.999 + 4.70i)T + (-17.3 - 7.72i)T^{2} \)
23 \( 1 + (6.52 - 3.76i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-1.66 - 7.81i)T + (-26.4 + 11.7i)T^{2} \)
31 \( 1 + (-3.77 + 2.73i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.57 - 7.38i)T + (-33.8 + 15.0i)T^{2} \)
41 \( 1 + (-3.46 + 3.84i)T + (-4.28 - 40.7i)T^{2} \)
43 \( 1 + (0.871 - 1.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-2.20 - 6.78i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (3.68 - 2.67i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-7.98 - 8.86i)T + (-6.16 + 58.6i)T^{2} \)
61 \( 1 + (3.76 + 8.46i)T + (-40.8 + 45.3i)T^{2} \)
67 \( 1 + (5.36 + 9.28i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-9.88 + 4.39i)T + (47.5 - 52.7i)T^{2} \)
73 \( 1 + (1.28 - 3.96i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (7.00 - 5.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (0.377 - 0.519i)T + (-25.6 - 78.9i)T^{2} \)
89 \( 1 + (3.48 - 2.01i)T + (44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.03 - 0.213i)T + (94.8 - 20.1i)T^{2} \)
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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

−10.97770684871033709269896848813, −9.814503083136937129890496027371, −9.169142073923010350513706856135, −8.402801473370920275981716907517, −7.32883530998046838920645185953, −6.50860364275141364587909590115, −5.97364784661761073933845656839, −4.30458488301323562874015053582, −3.22266574892726529679618613371, −1.35573389502768684942195738351, 0.76116763334026207428900711408, 2.25324463764189551745976325846, 3.65544301218128966519453663095, 4.56612541696350613445272940412, 5.99292788809835492256651981986, 7.31653379249638975840170461879, 8.146995566012186069856267235935, 8.561635856638089153778536874837, 9.835672886236096655854619773838, 10.27935983122362207577005544294

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