Properties

Label 2-531-59.58-c4-0-21
Degree $2$
Conductor $531$
Sign $-0.861 + 0.506i$
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  + 7.77i·2-s − 44.4·4-s − 28.5·5-s + 72.7·7-s − 221. i·8-s − 222. i·10-s + 9.33i·11-s − 106. i·13-s + 565. i·14-s + 1.01e3·16-s − 337.·17-s + 546.·19-s + 1.27e3·20-s − 72.6·22-s − 39.3i·23-s + ⋯
L(s)  = 1  + 1.94i·2-s − 2.78·4-s − 1.14·5-s + 1.48·7-s − 3.46i·8-s − 2.22i·10-s + 0.0771i·11-s − 0.631i·13-s + 2.88i·14-s + 3.95·16-s − 1.16·17-s + 1.51·19-s + 3.17·20-s − 0.150·22-s − 0.0742i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.861 + 0.506i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.861 + 0.506i$
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: \((2,\ 531,\ (\ :2),\ -0.861 + 0.506i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.108729046\)
\(L(\frac12)\) \(\approx\) \(1.108729046\)
\(L(3)\) 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 \( 1 + (-3.00e3 + 1.76e3i)T \)
good2 \( 1 - 7.77iT - 16T^{2} \)
5 \( 1 + 28.5T + 625T^{2} \)
7 \( 1 - 72.7T + 2.40e3T^{2} \)
11 \( 1 - 9.33iT - 1.46e4T^{2} \)
13 \( 1 + 106. iT - 2.85e4T^{2} \)
17 \( 1 + 337.T + 8.35e4T^{2} \)
19 \( 1 - 546.T + 1.30e5T^{2} \)
23 \( 1 + 39.3iT - 2.79e5T^{2} \)
29 \( 1 - 444.T + 7.07e5T^{2} \)
31 \( 1 - 1.23e3iT - 9.23e5T^{2} \)
37 \( 1 - 1.94e3iT - 1.87e6T^{2} \)
41 \( 1 - 835.T + 2.82e6T^{2} \)
43 \( 1 + 2.24e3iT - 3.41e6T^{2} \)
47 \( 1 - 3.91e3iT - 4.87e6T^{2} \)
53 \( 1 + 3.82e3T + 7.89e6T^{2} \)
61 \( 1 + 2.34e3iT - 1.38e7T^{2} \)
67 \( 1 + 3.65e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.63e3T + 2.54e7T^{2} \)
73 \( 1 + 8.92e3iT - 2.83e7T^{2} \)
79 \( 1 - 1.49e3T + 3.89e7T^{2} \)
83 \( 1 - 3.98e3iT - 4.74e7T^{2} \)
89 \( 1 - 2.53e3iT - 6.27e7T^{2} \)
97 \( 1 - 9.21e3iT - 8.85e7T^{2} \)
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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

−10.68231242669486584792733299541, −9.379528499663163692840103977719, −8.443680110782027858538692055932, −7.922779128205248318095135081464, −7.36920760424655264062317015497, −6.36240561372821936593293232407, −5.01204835220778833259710217552, −4.75029656159066041905775402934, −3.53489346001145562121005219565, −0.963506264450913973378672275150, 0.39283917695136928030351070182, 1.52222877039916514033468849736, 2.57841486964444615601978857545, 3.91663432393340475969410896776, 4.41878715811062747372776216900, 5.36416213147957076957711061113, 7.47228863771794987282159369868, 8.259139700596342685430606394766, 8.996252106611364249981577748326, 9.939018934199817710525221119912

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