Properties

Label 2-531-59.58-c4-0-62
Degree $2$
Conductor $531$
Sign $-0.317 + 0.948i$
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  − 2.43i·2-s + 10.0·4-s − 25.9·5-s − 17.4·7-s − 63.4i·8-s + 63.0i·10-s − 44.7i·11-s + 157. i·13-s + 42.4i·14-s + 7.22·16-s + 328.·17-s + 678.·19-s − 261.·20-s − 108.·22-s + 424. i·23-s + ⋯
L(s)  = 1  − 0.607i·2-s + 0.630·4-s − 1.03·5-s − 0.356·7-s − 0.991i·8-s + 0.630i·10-s − 0.369i·11-s + 0.931i·13-s + 0.216i·14-s + 0.0282·16-s + 1.13·17-s + 1.87·19-s − 0.654·20-s − 0.224·22-s + 0.802i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.317 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $-0.317 + 0.948i$
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: \((2,\ 531,\ (\ :2),\ -0.317 + 0.948i)\)

Particular Values

\(L(\frac{5}{2})\) \(\approx\) \(1.799361717\)
\(L(\frac12)\) \(\approx\) \(1.799361717\)
\(L(3)\) 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 \( 1 + (-1.10e3 + 3.30e3i)T \)
good2 \( 1 + 2.43iT - 16T^{2} \)
5 \( 1 + 25.9T + 625T^{2} \)
7 \( 1 + 17.4T + 2.40e3T^{2} \)
11 \( 1 + 44.7iT - 1.46e4T^{2} \)
13 \( 1 - 157. iT - 2.85e4T^{2} \)
17 \( 1 - 328.T + 8.35e4T^{2} \)
19 \( 1 - 678.T + 1.30e5T^{2} \)
23 \( 1 - 424. iT - 2.79e5T^{2} \)
29 \( 1 + 870.T + 7.07e5T^{2} \)
31 \( 1 + 1.38e3iT - 9.23e5T^{2} \)
37 \( 1 - 2.56e3iT - 1.87e6T^{2} \)
41 \( 1 + 503.T + 2.82e6T^{2} \)
43 \( 1 + 2.00e3iT - 3.41e6T^{2} \)
47 \( 1 + 2.42e3iT - 4.87e6T^{2} \)
53 \( 1 - 1.52e3T + 7.89e6T^{2} \)
61 \( 1 + 5.70e3iT - 1.38e7T^{2} \)
67 \( 1 + 4.77e3iT - 2.01e7T^{2} \)
71 \( 1 - 2.79e3T + 2.54e7T^{2} \)
73 \( 1 + 4.71e3iT - 2.83e7T^{2} \)
79 \( 1 - 8.66e3T + 3.89e7T^{2} \)
83 \( 1 + 7.45e3iT - 4.74e7T^{2} \)
89 \( 1 - 4.00e3iT - 6.27e7T^{2} \)
97 \( 1 + 5.91e3iT - 8.85e7T^{2} \)
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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

−9.935492272956244851815475643132, −9.430262543551522619922688960547, −7.941131987991074362799828824724, −7.41336584251958442126061896585, −6.41080971707428111337242492993, −5.23606698955176069476715734990, −3.65950527359728770996784224439, −3.33829696757752954554760053544, −1.77714585177937276747740984393, −0.53198668927452106658593853367, 1.05867625242863830082145671116, 2.80100455632460139762245635005, 3.65588266327453976664582505088, 5.15261661312582494900010523529, 5.89446586091208148233316539673, 7.24689362411878290450922261230, 7.52451536170323763599880244690, 8.393125203470068366478146388351, 9.632130122253162215586696828433, 10.56918746942799110074638981339

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