Properties

Label 2-531-1.1-c7-0-9
Degree $2$
Conductor $531$
Sign $1$
Analytic cond. $165.876$
Root an. cond. $12.8793$
Motivic weight $7$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.413·2-s − 127.·4-s − 231.·5-s + 302.·7-s + 105.·8-s + 95.4·10-s − 1.28e3·11-s − 1.48e4·13-s − 124.·14-s + 1.63e4·16-s + 2.11e4·17-s − 1.75e3·19-s + 2.95e4·20-s + 529.·22-s − 9.19e4·23-s − 2.46e4·25-s + 6.13e3·26-s − 3.86e4·28-s − 6.26e4·29-s − 1.03e5·31-s − 2.02e4·32-s − 8.74e3·34-s − 6.98e4·35-s − 3.79e4·37-s + 724.·38-s − 2.44e4·40-s + 4.56e4·41-s + ⋯
L(s)  = 1  − 0.0365·2-s − 0.998·4-s − 0.826·5-s + 0.332·7-s + 0.0729·8-s + 0.0301·10-s − 0.290·11-s − 1.87·13-s − 0.0121·14-s + 0.996·16-s + 1.04·17-s − 0.0587·19-s + 0.825·20-s + 0.0105·22-s − 1.57·23-s − 0.316·25-s + 0.0684·26-s − 0.332·28-s − 0.477·29-s − 0.622·31-s − 0.109·32-s − 0.0381·34-s − 0.275·35-s − 0.123·37-s + 0.00214·38-s − 0.0603·40-s + 0.103·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 531 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(531\)    =    \(3^{2} \cdot 59\)
Sign: $1$
Analytic conductor: \(165.876\)
Root analytic conductor: \(12.8793\)
Motivic weight: \(7\)
Rational: no
Arithmetic: yes
Character: $\chi_{531} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.1783620910\)
\(L(\frac12)\) \(\approx\) \(0.1783620910\)
\(L(\frac{9}{2})\) 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 \( 1 - 2.05e5T \)
good2 \( 1 + 0.413T + 128T^{2} \)
5 \( 1 + 231.T + 7.81e4T^{2} \)
7 \( 1 - 302.T + 8.23e5T^{2} \)
11 \( 1 + 1.28e3T + 1.94e7T^{2} \)
13 \( 1 + 1.48e4T + 6.27e7T^{2} \)
17 \( 1 - 2.11e4T + 4.10e8T^{2} \)
19 \( 1 + 1.75e3T + 8.93e8T^{2} \)
23 \( 1 + 9.19e4T + 3.40e9T^{2} \)
29 \( 1 + 6.26e4T + 1.72e10T^{2} \)
31 \( 1 + 1.03e5T + 2.75e10T^{2} \)
37 \( 1 + 3.79e4T + 9.49e10T^{2} \)
41 \( 1 - 4.56e4T + 1.94e11T^{2} \)
43 \( 1 + 6.37e5T + 2.71e11T^{2} \)
47 \( 1 + 2.29e5T + 5.06e11T^{2} \)
53 \( 1 - 1.07e6T + 1.17e12T^{2} \)
61 \( 1 + 2.75e6T + 3.14e12T^{2} \)
67 \( 1 + 3.11e6T + 6.06e12T^{2} \)
71 \( 1 + 4.01e6T + 9.09e12T^{2} \)
73 \( 1 + 5.86e6T + 1.10e13T^{2} \)
79 \( 1 - 4.62e6T + 1.92e13T^{2} \)
83 \( 1 + 9.51e6T + 2.71e13T^{2} \)
89 \( 1 - 6.95e6T + 4.42e13T^{2} \)
97 \( 1 + 9.56e6T + 8.07e13T^{2} \)
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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

−9.839814801165412814754207790147, −8.758454280562074490139650495802, −7.75174937822278022742279371014, −7.50499730567538414087065234021, −5.80931558631071015616251800043, −4.92136932866083670894578528777, −4.16604079315358593559687634483, −3.14910555507934320562452429943, −1.72555592867344389029845331279, −0.18236111938158182420682333919, 0.18236111938158182420682333919, 1.72555592867344389029845331279, 3.14910555507934320562452429943, 4.16604079315358593559687634483, 4.92136932866083670894578528777, 5.80931558631071015616251800043, 7.50499730567538414087065234021, 7.75174937822278022742279371014, 8.758454280562074490139650495802, 9.839814801165412814754207790147

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