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Properties

Label 2-531-59.58-c4-0-9
Degree $2$
Conductor $531$
Sign $0.985 - 0.169i$
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  − 6.76i·2-s − 29.7·4-s − 6.77·5-s + 45.8·7-s + 93.3i·8-s + 45.8i·10-s − 152. i·11-s + 138. i·13-s − 310. i·14-s + 155.·16-s − 61.1·17-s − 184.·19-s + 201.·20-s − 1.03e3·22-s − 55.4i·23-s + ⋯
L(s)  = 1  − 1.69i·2-s − 1.86·4-s − 0.271·5-s + 0.936·7-s + 1.45i·8-s + 0.458i·10-s − 1.25i·11-s + 0.819i·13-s − 1.58i·14-s + 0.605·16-s − 0.211·17-s − 0.509·19-s + 0.504·20-s − 2.13·22-s − 0.104i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $0.985 - 0.169i$
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: \((2,\ 531,\ (\ :2),\ 0.985 - 0.169i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(0.4920692731\)
\(L(\frac12)\) \(\approx\) \(0.4920692731\)
\(L(3)\) 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 \( 1 + (3.43e3 - 589. i)T \)
good2 \( 1 + 6.76iT - 16T^{2} \)
5 \( 1 + 6.77T + 625T^{2} \)
7 \( 1 - 45.8T + 2.40e3T^{2} \)
11 \( 1 + 152. iT - 1.46e4T^{2} \)
13 \( 1 - 138. iT - 2.85e4T^{2} \)
17 \( 1 + 61.1T + 8.35e4T^{2} \)
19 \( 1 + 184.T + 1.30e5T^{2} \)
23 \( 1 + 55.4iT - 2.79e5T^{2} \)
29 \( 1 + 290.T + 7.07e5T^{2} \)
31 \( 1 - 681. iT - 9.23e5T^{2} \)
37 \( 1 - 1.42e3iT - 1.87e6T^{2} \)
41 \( 1 + 1.53e3T + 2.82e6T^{2} \)
43 \( 1 + 404. iT - 3.41e6T^{2} \)
47 \( 1 - 1.60e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.32e3T + 7.89e6T^{2} \)
61 \( 1 + 2.73e3iT - 1.38e7T^{2} \)
67 \( 1 - 2.22e3iT - 2.01e7T^{2} \)
71 \( 1 - 6.15e3T + 2.54e7T^{2} \)
73 \( 1 - 2.81e3iT - 2.83e7T^{2} \)
79 \( 1 - 7.74e3T + 3.89e7T^{2} \)
83 \( 1 - 648. iT - 4.74e7T^{2} \)
89 \( 1 - 5.75e3iT - 6.27e7T^{2} \)
97 \( 1 + 1.51e3iT - 8.85e7T^{2} \)
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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

−10.61489025109836444943642785126, −9.561792283895242436007911407311, −8.691938354475954633788251782035, −8.039411324772208360444903470850, −6.53968270451938065868550338227, −5.15813947864652249041462592202, −4.22307671275955999920327973406, −3.34441395275032358642932964405, −2.13073878623780526986045201169, −1.17151393413301902345303215835, 0.13448164734183225066809298813, 2.00234118070164005634351751526, 3.99051648155684627005805495942, 4.84290484071568533267593421769, 5.62367697745946436336134391389, 6.69321460616566231777628274261, 7.62949950020865624076031958901, 7.995553739690594709311971184076, 8.996815448371369139546999910467, 9.927708970947314703606832363242

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