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Properties

Label 2-531-59.58-c4-0-17
Degree $2$
Conductor $531$
Sign $-0.995 - 0.0942i$
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  + 7.81i·2-s − 45.1·4-s − 21.3·5-s − 30.7·7-s − 227. i·8-s − 166. i·10-s − 207. i·11-s + 206. i·13-s − 240. i·14-s + 1.05e3·16-s + 391.·17-s − 321.·19-s + 962.·20-s + 1.62e3·22-s − 287. i·23-s + ⋯
L(s)  = 1  + 1.95i·2-s − 2.82·4-s − 0.852·5-s − 0.627·7-s − 3.55i·8-s − 1.66i·10-s − 1.71i·11-s + 1.21i·13-s − 1.22i·14-s + 4.13·16-s + 1.35·17-s − 0.890·19-s + 2.40·20-s + 3.35·22-s − 0.544i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.995 - 0.0942i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.995 - 0.0942i$
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: \((2,\ 531,\ (\ :2),\ -0.995 - 0.0942i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.6782517987\)
\(L(\frac12)\) \(\approx\) \(0.6782517987\)
\(L(3)\) 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 \( 1 + (-3.46e3 - 327. i)T \)
good2 \( 1 - 7.81iT - 16T^{2} \)
5 \( 1 + 21.3T + 625T^{2} \)
7 \( 1 + 30.7T + 2.40e3T^{2} \)
11 \( 1 + 207. iT - 1.46e4T^{2} \)
13 \( 1 - 206. iT - 2.85e4T^{2} \)
17 \( 1 - 391.T + 8.35e4T^{2} \)
19 \( 1 + 321.T + 1.30e5T^{2} \)
23 \( 1 + 287. iT - 2.79e5T^{2} \)
29 \( 1 + 1.02e3T + 7.07e5T^{2} \)
31 \( 1 - 560. iT - 9.23e5T^{2} \)
37 \( 1 + 1.25e3iT - 1.87e6T^{2} \)
41 \( 1 - 1.52e3T + 2.82e6T^{2} \)
43 \( 1 + 2.15e3iT - 3.41e6T^{2} \)
47 \( 1 - 245. iT - 4.87e6T^{2} \)
53 \( 1 - 1.14e3T + 7.89e6T^{2} \)
61 \( 1 + 518. iT - 1.38e7T^{2} \)
67 \( 1 - 2.49e3iT - 2.01e7T^{2} \)
71 \( 1 + 9.31e3T + 2.54e7T^{2} \)
73 \( 1 - 8.00e3iT - 2.83e7T^{2} \)
79 \( 1 - 5.13e3T + 3.89e7T^{2} \)
83 \( 1 + 1.34e4iT - 4.74e7T^{2} \)
89 \( 1 - 3.14e3iT - 6.27e7T^{2} \)
97 \( 1 - 5.32e3iT - 8.85e7T^{2} \)
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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

−10.39125590947268415653339235871, −9.223905087881045135115818521155, −8.663138455192379563532404534841, −7.84899667508357295184785427151, −7.04963612275318627138898105427, −6.15990818956370466209055047837, −5.49641381240288242373004128703, −4.12883028216502057421509066197, −3.53662889196341875108127013686, −0.61485511977552516471593476889, 0.33176932404686046710915504213, 1.65632428728655744587224877212, 2.87534168011470932467604003525, 3.72762183577280430953796228509, 4.55163430123180778568657249741, 5.65262538448136654496401299866, 7.54037167433932495254254830368, 8.180283955160574290273858907292, 9.549871201355333886328011038293, 9.860367187288364106871658369758

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