Properties

Label 2-531-3.2-c4-0-40
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  + 6.38i·2-s − 24.7·4-s + 16.4i·5-s + 84.4·7-s − 55.9i·8-s − 104.·10-s − 31.3i·11-s + 43.8·13-s + 539. i·14-s − 39.1·16-s − 243. i·17-s + 602.·19-s − 406. i·20-s + 199.·22-s − 401. i·23-s + ⋯
L(s)  = 1  + 1.59i·2-s − 1.54·4-s + 0.657i·5-s + 1.72·7-s − 0.873i·8-s − 1.04·10-s − 0.258i·11-s + 0.259·13-s + 2.75i·14-s − 0.153·16-s − 0.841i·17-s + 1.66·19-s − 1.01i·20-s + 0.413·22-s − 0.758i·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

\(L(\frac{5}{2})\) \(\approx\) \(2.707762598\)
\(L(\frac12)\) \(\approx\) \(2.707762598\)
\(L(3)\) 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 \( 1 - 6.38iT - 16T^{2} \)
5 \( 1 - 16.4iT - 625T^{2} \)
7 \( 1 - 84.4T + 2.40e3T^{2} \)
11 \( 1 + 31.3iT - 1.46e4T^{2} \)
13 \( 1 - 43.8T + 2.85e4T^{2} \)
17 \( 1 + 243. iT - 8.35e4T^{2} \)
19 \( 1 - 602.T + 1.30e5T^{2} \)
23 \( 1 + 401. iT - 2.79e5T^{2} \)
29 \( 1 - 1.57e3iT - 7.07e5T^{2} \)
31 \( 1 - 306.T + 9.23e5T^{2} \)
37 \( 1 - 2.50e3T + 1.87e6T^{2} \)
41 \( 1 + 1.68e3iT - 2.82e6T^{2} \)
43 \( 1 - 2.97e3T + 3.41e6T^{2} \)
47 \( 1 + 890. iT - 4.87e6T^{2} \)
53 \( 1 - 4.49e3iT - 7.89e6T^{2} \)
61 \( 1 + 4.99e3T + 1.38e7T^{2} \)
67 \( 1 + 5.52e3T + 2.01e7T^{2} \)
71 \( 1 + 6.51e3iT - 2.54e7T^{2} \)
73 \( 1 - 1.30e3T + 2.83e7T^{2} \)
79 \( 1 + 7.10e3T + 3.89e7T^{2} \)
83 \( 1 - 7.73e3iT - 4.74e7T^{2} \)
89 \( 1 + 9.38e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.68e4T + 8.85e7T^{2} \)
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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

−10.72593537230354649189478607192, −9.255438444660924578025305824371, −8.572193723426590795536468400875, −7.51955487153273140497968655249, −7.29821549107505314368821194294, −6.03362328972085207443281175284, −5.18474400911905035929013051073, −4.46099406095042747914890085439, −2.79950733898324913991985103198, −1.06961512223574647406646129808, 0.950647312262978150945810779572, 1.52207302755826796076918710834, 2.69752383363375753682551473305, 4.11639884053370584557116128100, 4.70942095220942345055938458553, 5.77095364532825272804690477295, 7.60925796938174672858367774902, 8.281689342722499273272026052395, 9.293516777362704605559816393339, 9.982889688952851980906867179855

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