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Properties

Label 2-531-59.58-c4-0-10
Degree $2$
Conductor $531$
Sign $0.486 + 0.873i$
Analytic cond. $54.8894$
Root an. cond. $7.40874$
Motivic weight $4$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.65i·2-s − 42.5·4-s + 38.1·5-s − 35.1·7-s − 203. i·8-s + 292. i·10-s + 147. i·11-s − 17.0i·13-s − 268. i·14-s + 875.·16-s + 354.·17-s − 647.·19-s − 1.62e3·20-s − 1.13e3·22-s + 862. i·23-s + ⋯
L(s)  = 1  + 1.91i·2-s − 2.66·4-s + 1.52·5-s − 0.717·7-s − 3.17i·8-s + 2.92i·10-s + 1.22i·11-s − 0.101i·13-s − 1.37i·14-s + 3.41·16-s + 1.22·17-s − 1.79·19-s − 4.06·20-s − 2.33·22-s + 1.63i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (235, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6325551513\)
\(L(\frac12)\) \(\approx\) \(0.6325551513\)
\(L(3)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
59 \( 1 + (1.69e3 + 3.04e3i)T \)
good2 \( 1 - 7.65iT - 16T^{2} \)
5 \( 1 - 38.1T + 625T^{2} \)
7 \( 1 + 35.1T + 2.40e3T^{2} \)
11 \( 1 - 147. iT - 1.46e4T^{2} \)
13 \( 1 + 17.0iT - 2.85e4T^{2} \)
17 \( 1 - 354.T + 8.35e4T^{2} \)
19 \( 1 + 647.T + 1.30e5T^{2} \)
23 \( 1 - 862. iT - 2.79e5T^{2} \)
29 \( 1 + 440.T + 7.07e5T^{2} \)
31 \( 1 - 347. iT - 9.23e5T^{2} \)
37 \( 1 - 972. iT - 1.87e6T^{2} \)
41 \( 1 + 2.78e3T + 2.82e6T^{2} \)
43 \( 1 + 2.83e3iT - 3.41e6T^{2} \)
47 \( 1 + 1.28e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.10e3T + 7.89e6T^{2} \)
61 \( 1 + 5.98e3iT - 1.38e7T^{2} \)
67 \( 1 - 3.35e3iT - 2.01e7T^{2} \)
71 \( 1 - 1.39e3T + 2.54e7T^{2} \)
73 \( 1 + 1.64e3iT - 2.83e7T^{2} \)
79 \( 1 + 8.35e3T + 3.89e7T^{2} \)
83 \( 1 - 5.02e3iT - 4.74e7T^{2} \)
89 \( 1 + 5.29e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.40e4iT - 8.85e7T^{2} \)
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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

−10.23722997939043731504837582662, −9.818970365294608147051447175308, −9.150183268741287544871329540239, −8.152058164588237961222619783995, −7.05666871280884012686056245070, −6.49777090310419466088972090849, −5.62220160941991399630461685034, −5.00242000731662225265909815759, −3.63429979425244519414711478920, −1.73188840465135401589852494582, 0.16118922392220357221513396658, 1.35418288681372882999540436693, 2.42271649497781033529301240267, 3.18172880397379475049615700892, 4.37574373850149296398609521267, 5.62249182749547917522844023875, 6.31328520802812339253908010955, 8.363473831046418186194753195853, 9.005673027503382388124245087367, 9.852789301798440796800489698786

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