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Properties

Label 2-531-3.2-c4-0-31
Degree $2$
Conductor $531$
Sign $-0.816 - 0.577i$
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  + 4.16i·2-s − 1.34·4-s − 3.12i·5-s + 44.2·7-s + 61.0i·8-s + 13.0·10-s + 187. i·11-s + 125.·13-s + 184. i·14-s − 275.·16-s + 44.6i·17-s + 38.8·19-s + 4.19i·20-s − 781.·22-s − 316. i·23-s + ⋯
L(s)  = 1  + 1.04i·2-s − 0.0840·4-s − 0.124i·5-s + 0.902·7-s + 0.953i·8-s + 0.130·10-s + 1.55i·11-s + 0.741·13-s + 0.939i·14-s − 1.07·16-s + 0.154i·17-s + 0.107·19-s + 0.0104i·20-s − 1.61·22-s − 0.598i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.816 - 0.577i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.816 - 0.577i$
Analytic conductor: \(54.8894\)
Root analytic conductor: \(7.40874\)
Motivic weight: \(4\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (296, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :2),\ -0.816 - 0.577i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.651819489\)
\(L(\frac12)\) \(\approx\) \(2.651819489\)
\(L(3)\) 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 \( 1 + 453. iT \)
good2 \( 1 - 4.16iT - 16T^{2} \)
5 \( 1 + 3.12iT - 625T^{2} \)
7 \( 1 - 44.2T + 2.40e3T^{2} \)
11 \( 1 - 187. iT - 1.46e4T^{2} \)
13 \( 1 - 125.T + 2.85e4T^{2} \)
17 \( 1 - 44.6iT - 8.35e4T^{2} \)
19 \( 1 - 38.8T + 1.30e5T^{2} \)
23 \( 1 + 316. iT - 2.79e5T^{2} \)
29 \( 1 + 871. iT - 7.07e5T^{2} \)
31 \( 1 - 1.16e3T + 9.23e5T^{2} \)
37 \( 1 + 626.T + 1.87e6T^{2} \)
41 \( 1 - 1.55e3iT - 2.82e6T^{2} \)
43 \( 1 + 429.T + 3.41e6T^{2} \)
47 \( 1 - 3.83e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.29e3iT - 7.89e6T^{2} \)
61 \( 1 - 1.48e3T + 1.38e7T^{2} \)
67 \( 1 + 1.37e3T + 2.01e7T^{2} \)
71 \( 1 - 3.11e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.00e3T + 2.83e7T^{2} \)
79 \( 1 - 1.54e3T + 3.89e7T^{2} \)
83 \( 1 - 7.95e3iT - 4.74e7T^{2} \)
89 \( 1 - 6.75e3iT - 6.27e7T^{2} \)
97 \( 1 - 1.24e4T + 8.85e7T^{2} \)
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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

−10.61648345812827650061631243985, −9.563737205304634846294521243948, −8.418080153042535136947071112226, −7.910870555000783828351207374508, −6.94382160195065936660910448573, −6.19909854184930343808887849300, −5.00861096198340295005709799485, −4.39849016584915751708484182301, −2.54310023699752356813363037632, −1.39241993297759103211571850959, 0.70966709220607999980713248254, 1.62043282042542063024114470670, 2.95976673882775754353145896056, 3.70664737025738629265354662453, 5.03986224582994297006813541140, 6.15146129033913988348667522352, 7.16952565355951142646294999909, 8.380539899976877837202818934221, 8.983874256511903005927752786411, 10.31115088119163232751320940537

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