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Properties

Label 2-531-3.2-c4-0-25
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  + 4.32i·2-s − 2.68·4-s + 35.4i·5-s + 80.8·7-s + 57.5i·8-s − 153.·10-s + 74.8i·11-s − 179.·13-s + 349. i·14-s − 291.·16-s − 35.1i·17-s − 52.2·19-s − 94.9i·20-s − 323.·22-s − 217. i·23-s + ⋯
L(s)  = 1  + 1.08i·2-s − 0.167·4-s + 1.41i·5-s + 1.65·7-s + 0.899i·8-s − 1.53·10-s + 0.618i·11-s − 1.05·13-s + 1.78i·14-s − 1.13·16-s − 0.121i·17-s − 0.144·19-s − 0.237i·20-s − 0.668·22-s − 0.410i·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

\(L(\frac{5}{2})\) \(\approx\) \(2.236215966\)
\(L(\frac12)\) \(\approx\) \(2.236215966\)
\(L(3)\) 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 \( 1 - 4.32iT - 16T^{2} \)
5 \( 1 - 35.4iT - 625T^{2} \)
7 \( 1 - 80.8T + 2.40e3T^{2} \)
11 \( 1 - 74.8iT - 1.46e4T^{2} \)
13 \( 1 + 179.T + 2.85e4T^{2} \)
17 \( 1 + 35.1iT - 8.35e4T^{2} \)
19 \( 1 + 52.2T + 1.30e5T^{2} \)
23 \( 1 + 217. iT - 2.79e5T^{2} \)
29 \( 1 - 557. iT - 7.07e5T^{2} \)
31 \( 1 - 342.T + 9.23e5T^{2} \)
37 \( 1 + 940.T + 1.87e6T^{2} \)
41 \( 1 - 2.60e3iT - 2.82e6T^{2} \)
43 \( 1 - 200.T + 3.41e6T^{2} \)
47 \( 1 + 250. iT - 4.87e6T^{2} \)
53 \( 1 - 763. iT - 7.89e6T^{2} \)
61 \( 1 - 5.63e3T + 1.38e7T^{2} \)
67 \( 1 + 937.T + 2.01e7T^{2} \)
71 \( 1 + 6.90e3iT - 2.54e7T^{2} \)
73 \( 1 + 4.21e3T + 2.83e7T^{2} \)
79 \( 1 + 672.T + 3.89e7T^{2} \)
83 \( 1 + 3.21e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.05e4iT - 6.27e7T^{2} \)
97 \( 1 - 4.42e3T + 8.85e7T^{2} \)
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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

−10.84246215556914897490504440879, −9.979220543669452434179203664229, −8.613378256643021981883482850974, −7.73444864665269369975170265681, −7.21645809656806911530758634450, −6.47012424445546455984057022130, −5.27270982646929650936581044228, −4.52879315357362038590599373881, −2.76613346490669565624632685212, −1.86057865488906966224704629019, 0.54268149505021888879072856538, 1.46986046479805068456011662441, 2.34780203034621974114444766993, 3.94152338413274297792924856538, 4.79039890477409710094521130849, 5.57542869077624644248517968883, 7.20472233091079400653353186657, 8.193193942696529366633358720646, 8.869448290329823505966661832766, 9.845305154921696728144799777403

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