Properties

Label 2-531-59.58-c4-0-46
Degree $2$
Conductor $531$
Sign $-0.999 - 0.0296i$
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  + 6.70i·2-s − 29.0·4-s + 41.0·5-s − 6.70·7-s − 87.3i·8-s + 275. i·10-s − 21.2i·11-s + 73.2i·13-s − 45.0i·14-s + 121.·16-s − 76.4·17-s + 439.·19-s − 1.19e3·20-s + 142.·22-s − 164. i·23-s + ⋯
L(s)  = 1  + 1.67i·2-s − 1.81·4-s + 1.64·5-s − 0.136·7-s − 1.36i·8-s + 2.75i·10-s − 0.175i·11-s + 0.433i·13-s − 0.229i·14-s + 0.476·16-s − 0.264·17-s + 1.21·19-s − 2.98·20-s + 0.295·22-s − 0.311i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0296i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0296i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.999 - 0.0296i$
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: \((2,\ 531,\ (\ :2),\ -0.999 - 0.0296i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.434123061\)
\(L(\frac12)\) \(\approx\) \(2.434123061\)
\(L(3)\) 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 \( 1 + (-3.47e3 - 103. i)T \)
good2 \( 1 - 6.70iT - 16T^{2} \)
5 \( 1 - 41.0T + 625T^{2} \)
7 \( 1 + 6.70T + 2.40e3T^{2} \)
11 \( 1 + 21.2iT - 1.46e4T^{2} \)
13 \( 1 - 73.2iT - 2.85e4T^{2} \)
17 \( 1 + 76.4T + 8.35e4T^{2} \)
19 \( 1 - 439.T + 1.30e5T^{2} \)
23 \( 1 + 164. iT - 2.79e5T^{2} \)
29 \( 1 - 788.T + 7.07e5T^{2} \)
31 \( 1 - 754. iT - 9.23e5T^{2} \)
37 \( 1 - 1.70e3iT - 1.87e6T^{2} \)
41 \( 1 + 72.3T + 2.82e6T^{2} \)
43 \( 1 - 2.69e3iT - 3.41e6T^{2} \)
47 \( 1 - 2.90e3iT - 4.87e6T^{2} \)
53 \( 1 + 1.17e3T + 7.89e6T^{2} \)
61 \( 1 - 2.04e3iT - 1.38e7T^{2} \)
67 \( 1 - 796. iT - 2.01e7T^{2} \)
71 \( 1 - 4.86e3T + 2.54e7T^{2} \)
73 \( 1 - 1.82e3iT - 2.83e7T^{2} \)
79 \( 1 + 7.08e3T + 3.89e7T^{2} \)
83 \( 1 + 4.42e3iT - 4.74e7T^{2} \)
89 \( 1 + 8.04e3iT - 6.27e7T^{2} \)
97 \( 1 + 2.35e3iT - 8.85e7T^{2} \)
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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

−10.20498637851083067706970854147, −9.524058125788799011806845256303, −8.839291064600661796665165415211, −7.87988482909949158716975395490, −6.70772057437179264670075207660, −6.30109803838491911698359338683, −5.37301323838136782034514698884, −4.64851236561140060220313880430, −2.84256827836518700681732359918, −1.28343044164461353504178487074, 0.65861238985265980643337822198, 1.77822916643467119732285665222, 2.54944385633942153530406151037, 3.59730555125863468990028474203, 4.96830704281953342815781059611, 5.75747417821251430712865911704, 6.99491311739934605399434497026, 8.523766444046212731309336458606, 9.491884034680119902149981188287, 9.853942100419603917318739626706

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