Properties

Label 2-531-59.58-c4-0-33
Degree $2$
Conductor $531$
Sign $-0.345 - 0.938i$
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  + 0.850i·2-s + 15.2·4-s + 11.2·5-s − 26.4·7-s + 26.5i·8-s + 9.58i·10-s + 19.9i·11-s + 190. i·13-s − 22.4i·14-s + 221.·16-s + 159.·17-s − 294.·19-s + 172.·20-s − 17.0·22-s − 165. i·23-s + ⋯
L(s)  = 1  + 0.212i·2-s + 0.954·4-s + 0.450·5-s − 0.539·7-s + 0.415i·8-s + 0.0958i·10-s + 0.165i·11-s + 1.12i·13-s − 0.114i·14-s + 0.866·16-s + 0.551·17-s − 0.814·19-s + 0.430·20-s − 0.0351·22-s − 0.312i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.345 - 0.938i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.345 - 0.938i$
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: \((2,\ 531,\ (\ :2),\ -0.345 - 0.938i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(2.149702935\)
\(L(\frac12)\) \(\approx\) \(2.149702935\)
\(L(3)\) 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 \( 1 + (-1.20e3 - 3.26e3i)T \)
good2 \( 1 - 0.850iT - 16T^{2} \)
5 \( 1 - 11.2T + 625T^{2} \)
7 \( 1 + 26.4T + 2.40e3T^{2} \)
11 \( 1 - 19.9iT - 1.46e4T^{2} \)
13 \( 1 - 190. iT - 2.85e4T^{2} \)
17 \( 1 - 159.T + 8.35e4T^{2} \)
19 \( 1 + 294.T + 1.30e5T^{2} \)
23 \( 1 + 165. iT - 2.79e5T^{2} \)
29 \( 1 + 513.T + 7.07e5T^{2} \)
31 \( 1 - 1.62e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.77e3iT - 1.87e6T^{2} \)
41 \( 1 - 2.06e3T + 2.82e6T^{2} \)
43 \( 1 - 2.18e3iT - 3.41e6T^{2} \)
47 \( 1 + 4.25e3iT - 4.87e6T^{2} \)
53 \( 1 + 246.T + 7.89e6T^{2} \)
61 \( 1 + 1.49e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.88e3iT - 2.01e7T^{2} \)
71 \( 1 + 1.86e3T + 2.54e7T^{2} \)
73 \( 1 - 619. iT - 2.83e7T^{2} \)
79 \( 1 - 2.37e3T + 3.89e7T^{2} \)
83 \( 1 - 4.60e3iT - 4.74e7T^{2} \)
89 \( 1 - 8.91e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.37e3iT - 8.85e7T^{2} \)
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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

−10.44982225417584145691552128637, −9.735576618803864668385106856657, −8.735776413664171134192373275658, −7.67606754027548422941422594749, −6.67292410640000625327918590302, −6.22420926487248743687742416241, −5.04304739991793005345988050229, −3.66293201893199267657536204476, −2.45735632835045047912925753610, −1.47373042341553110901926494703, 0.49541987702560929636069595269, 1.93618687756950130535601724151, 2.91110877463652957767226851636, 3.95771434613406292415059870533, 5.73262598717287906219887336964, 6.03528186696741975548373591357, 7.32061456227683046203853278620, 7.980645612069227410608092320746, 9.343714769925615999858461221483, 10.04958436576268373227053460870

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