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Properties

Label 2-572-13.9-c1-0-3
Degree $2$
Conductor $572$
Sign $0.938 - 0.346i$
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  + (0.194 − 0.336i)3-s + 2.23·5-s + (0.258 + 0.447i)7-s + (1.42 + 2.46i)9-s + (−0.5 + 0.866i)11-s + (0.353 + 3.58i)13-s + (0.434 − 0.752i)15-s + (0.907 + 1.57i)17-s + (−1.34 − 2.33i)19-s + 0.200·21-s + (2.70 − 4.68i)23-s − 0.00768·25-s + 2.27·27-s + (0.181 − 0.314i)29-s + 4.11·31-s + ⋯
L(s)  = 1  + (0.112 − 0.194i)3-s + 0.999·5-s + (0.0976 + 0.169i)7-s + (0.474 + 0.822i)9-s + (−0.150 + 0.261i)11-s + (0.0981 + 0.995i)13-s + (0.112 − 0.194i)15-s + (0.220 + 0.381i)17-s + (−0.309 − 0.535i)19-s + 0.0438·21-s + (0.564 − 0.977i)23-s − 0.00153·25-s + 0.437·27-s + (0.0337 − 0.0583i)29-s + 0.739·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 572 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.938 - 0.346i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(572\)    =    \(2^{2} \cdot 11 \cdot 13\)
Sign: $0.938 - 0.346i$
Analytic conductor: \(4.56744\)
Root analytic conductor: \(2.13715\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{572} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 572,\ (\ :1/2),\ 0.938 - 0.346i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.81850 + 0.325017i\)
\(L(\frac12)\) \(\approx\) \(1.81850 + 0.325017i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.353 - 3.58i)T \)
good3 \( 1 + (-0.194 + 0.336i)T + (-1.5 - 2.59i)T^{2} \)
5 \( 1 - 2.23T + 5T^{2} \)
7 \( 1 + (-0.258 - 0.447i)T + (-3.5 + 6.06i)T^{2} \)
17 \( 1 + (-0.907 - 1.57i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.34 + 2.33i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-2.70 + 4.68i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.181 + 0.314i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 - 4.11T + 31T^{2} \)
37 \( 1 + (0.813 - 1.40i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (0.518 - 0.897i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.382 + 0.663i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + 1.64T + 47T^{2} \)
53 \( 1 - 2.30T + 53T^{2} \)
59 \( 1 + (-1.37 - 2.38i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-0.688 - 1.19i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.08 + 8.80i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + (3.80 + 6.59i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + 1.43T + 73T^{2} \)
79 \( 1 + 9.23T + 79T^{2} \)
83 \( 1 + 2.93T + 83T^{2} \)
89 \( 1 + (-1.59 + 2.75i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (6.14 + 10.6i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.60378131028087325718446810851, −9.983838592238776629924522140962, −9.045170927437102479068369949891, −8.216838686380156147239244523370, −7.07372674805717496411538471129, −6.32723091153453004797804974257, −5.19359617686523160804343964748, −4.31398667356961008801616861528, −2.57176420848329672335944414495, −1.67562102768265580004421372325, 1.22673617364704207996158378862, 2.80546765595360708636522511415, 3.91722936658155364017480162080, 5.27471525592572887116186999988, 6.00842309681029085426564954549, 7.03584072290222540502437175061, 8.080745013654142215391112803340, 9.109254358376450554372196833775, 9.877044506995757780390484529866, 10.41720200671473122905197071879

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