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Properties

Label 2-572-44.43-c1-0-64
Degree $2$
Conductor $572$
Sign $-0.990 - 0.135i$
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  + (−0.936 + 1.05i)2-s − 2.78i·3-s + (−0.246 − 1.98i)4-s + 1.09·5-s + (2.95 + 2.60i)6-s − 3.18·7-s + (2.33 + 1.59i)8-s − 4.76·9-s + (−1.03 + 1.16i)10-s + (−3.20 − 0.849i)11-s + (−5.53 + 0.686i)12-s − i·13-s + (2.98 − 3.37i)14-s − 3.06i·15-s + (−3.87 + 0.977i)16-s − 0.404i·17-s + ⋯
L(s)  = 1  + (−0.662 + 0.749i)2-s − 1.60i·3-s + (−0.123 − 0.992i)4-s + 0.491·5-s + (1.20 + 1.06i)6-s − 1.20·7-s + (0.825 + 0.564i)8-s − 1.58·9-s + (−0.325 + 0.368i)10-s + (−0.966 − 0.256i)11-s + (−1.59 + 0.198i)12-s − 0.277i·13-s + (0.797 − 0.902i)14-s − 0.791i·15-s + (−0.969 + 0.244i)16-s − 0.0979i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $-0.990 - 0.135i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (131, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ -0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0190146 + 0.279907i\)
\(L(\frac12)\) \(\approx\) \(0.0190146 + 0.279907i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.936 - 1.05i)T \)
11 \( 1 + (3.20 + 0.849i)T \)
13 \( 1 + iT \)
good3 \( 1 + 2.78iT - 3T^{2} \)
5 \( 1 - 1.09T + 5T^{2} \)
7 \( 1 + 3.18T + 7T^{2} \)
17 \( 1 + 0.404iT - 17T^{2} \)
19 \( 1 - 0.124T + 19T^{2} \)
23 \( 1 - 4.72iT - 23T^{2} \)
29 \( 1 + 9.08iT - 29T^{2} \)
31 \( 1 - 10.4iT - 31T^{2} \)
37 \( 1 - 2.86T + 37T^{2} \)
41 \( 1 - 3.30iT - 41T^{2} \)
43 \( 1 + 7.57T + 43T^{2} \)
47 \( 1 - 4.17iT - 47T^{2} \)
53 \( 1 + 11.5T + 53T^{2} \)
59 \( 1 + 6.68iT - 59T^{2} \)
61 \( 1 - 1.02iT - 61T^{2} \)
67 \( 1 + 11.3iT - 67T^{2} \)
71 \( 1 + 12.4iT - 71T^{2} \)
73 \( 1 - 2.91iT - 73T^{2} \)
79 \( 1 - 10.8T + 79T^{2} \)
83 \( 1 - 3.91T + 83T^{2} \)
89 \( 1 + 4.50T + 89T^{2} \)
97 \( 1 - 3.28T + 97T^{2} \)
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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

−9.987308338479070478935194673798, −9.352196459862846597413501998696, −8.099796224595379577241520069406, −7.68758285078746615585797754055, −6.57840128665828389242916207193, −6.18306435117245071868482355387, −5.23180689529596468368305160535, −2.99033780780447286461470191907, −1.72104664685398003272654575864, −0.18587424916637434775308541696, 2.44795935582761726236797529413, 3.41066541101589102789075930267, 4.33920620519209591624528082549, 5.44759370207533701879494280663, 6.70070128779495788840816834264, 8.035772737712595797271943720120, 9.086821771140374345205676531220, 9.614886342977703408305949089302, 10.22811786193942831869561729020, 10.72961107123381286807608054932

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