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Properties

Label 2-531-3.2-c4-0-19
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  − 6.40i·2-s − 25.0·4-s + 18.5i·5-s + 29.7·7-s + 57.8i·8-s + 118.·10-s + 32.7i·11-s − 137.·13-s − 190. i·14-s − 29.9·16-s − 443. i·17-s − 630.·19-s − 464. i·20-s + 209.·22-s + 445. i·23-s + ⋯
L(s)  = 1  − 1.60i·2-s − 1.56·4-s + 0.741i·5-s + 0.606·7-s + 0.903i·8-s + 1.18·10-s + 0.270i·11-s − 0.814·13-s − 0.971i·14-s − 0.116·16-s − 1.53i·17-s − 1.74·19-s − 1.16i·20-s + 0.433·22-s + 0.842i·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

\(L(\frac{5}{2})\) \(\approx\) \(1.475804906\)
\(L(\frac12)\) \(\approx\) \(1.475804906\)
\(L(3)\) 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 \( 1 + 6.40iT - 16T^{2} \)
5 \( 1 - 18.5iT - 625T^{2} \)
7 \( 1 - 29.7T + 2.40e3T^{2} \)
11 \( 1 - 32.7iT - 1.46e4T^{2} \)
13 \( 1 + 137.T + 2.85e4T^{2} \)
17 \( 1 + 443. iT - 8.35e4T^{2} \)
19 \( 1 + 630.T + 1.30e5T^{2} \)
23 \( 1 - 445. iT - 2.79e5T^{2} \)
29 \( 1 - 606. iT - 7.07e5T^{2} \)
31 \( 1 - 1.62e3T + 9.23e5T^{2} \)
37 \( 1 - 873.T + 1.87e6T^{2} \)
41 \( 1 - 1.86e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.79e3T + 3.41e6T^{2} \)
47 \( 1 + 1.71e3iT - 4.87e6T^{2} \)
53 \( 1 - 3.46e3iT - 7.89e6T^{2} \)
61 \( 1 - 4.80e3T + 1.38e7T^{2} \)
67 \( 1 + 3.86e3T + 2.01e7T^{2} \)
71 \( 1 - 9.17e3iT - 2.54e7T^{2} \)
73 \( 1 + 3.34e3T + 2.83e7T^{2} \)
79 \( 1 - 7.87e3T + 3.89e7T^{2} \)
83 \( 1 + 1.05e3iT - 4.74e7T^{2} \)
89 \( 1 - 1.33e4iT - 6.27e7T^{2} \)
97 \( 1 + 1.57e3T + 8.85e7T^{2} \)
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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

−10.32190070009214040931487457657, −9.645682770402645833849847215635, −8.718644297827532288808312426102, −7.52114325263386240070996999055, −6.57748591687506105909357534900, −4.97980274302550467812421572540, −4.24833412134371582090084668204, −2.87962704215770917953028144555, −2.32108229274920296873861505020, −0.965188069978366396923144329511, 0.45689227585342949267822526147, 2.16751205163286059429832446771, 4.29185164877623735588786940589, 4.74899395494935682943088843649, 5.93898130039711622893585880787, 6.53268591682564012296986665897, 7.76281713867678228447166325984, 8.384344832173840353895152215759, 8.875608621453530909005578720083, 10.13662188403137336604901770654

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