Properties

Label 2-531-59.58-c4-0-4
Degree $2$
Conductor $531$
Sign $0.669 - 0.742i$
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  − 4.64i·2-s − 5.59·4-s + 0.691·5-s − 76.1·7-s − 48.3i·8-s − 3.21i·10-s − 74.8i·11-s − 105. i·13-s + 353. i·14-s − 314.·16-s + 141.·17-s − 170.·19-s − 3.87·20-s − 347.·22-s − 226. i·23-s + ⋯
L(s)  = 1  − 1.16i·2-s − 0.349·4-s + 0.0276·5-s − 1.55·7-s − 0.755i·8-s − 0.0321i·10-s − 0.618i·11-s − 0.623i·13-s + 1.80i·14-s − 1.22·16-s + 0.491·17-s − 0.473·19-s − 0.00967·20-s − 0.718·22-s − 0.428i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.669 - 0.742i$
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: \((2,\ 531,\ (\ :2),\ 0.669 - 0.742i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.1123545481\)
\(L(\frac12)\) \(\approx\) \(0.1123545481\)
\(L(3)\) 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 \( 1 + (2.33e3 - 2.58e3i)T \)
good2 \( 1 + 4.64iT - 16T^{2} \)
5 \( 1 - 0.691T + 625T^{2} \)
7 \( 1 + 76.1T + 2.40e3T^{2} \)
11 \( 1 + 74.8iT - 1.46e4T^{2} \)
13 \( 1 + 105. iT - 2.85e4T^{2} \)
17 \( 1 - 141.T + 8.35e4T^{2} \)
19 \( 1 + 170.T + 1.30e5T^{2} \)
23 \( 1 + 226. iT - 2.79e5T^{2} \)
29 \( 1 - 677.T + 7.07e5T^{2} \)
31 \( 1 + 114. iT - 9.23e5T^{2} \)
37 \( 1 - 488. iT - 1.87e6T^{2} \)
41 \( 1 - 1.82e3T + 2.82e6T^{2} \)
43 \( 1 - 527. iT - 3.41e6T^{2} \)
47 \( 1 - 1.90e3iT - 4.87e6T^{2} \)
53 \( 1 + 4.68e3T + 7.89e6T^{2} \)
61 \( 1 - 6.41e3iT - 1.38e7T^{2} \)
67 \( 1 + 1.77e3iT - 2.01e7T^{2} \)
71 \( 1 + 170.T + 2.54e7T^{2} \)
73 \( 1 - 8.99e3iT - 2.83e7T^{2} \)
79 \( 1 + 204.T + 3.89e7T^{2} \)
83 \( 1 + 1.39e3iT - 4.74e7T^{2} \)
89 \( 1 + 3.00e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.22e3iT - 8.85e7T^{2} \)
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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

−10.32758319884334692191419113476, −9.799089414351025663851682463457, −8.953338156012618844468059151491, −7.72007471867307508180995155827, −6.52692963867363625740995895547, −5.89062565175119803562748396457, −4.23010028341207289042705608175, −3.23744823888220944127853057295, −2.62379479601978052633880517104, −1.03847707071242148978072173604, 0.03117198695788596035949288951, 2.06954450938057441156449136407, 3.38485459274211359141912911886, 4.64400304553638932165011727281, 5.86603113488134300279654027707, 6.48155224745483948980415429465, 7.21444446944273092387329164024, 8.101835582557694223558836331395, 9.246920760095729921887482515271, 9.796173896623718745585681364378

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