Properties

Label 2-531-1.1-c7-0-45
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  + 17.8·2-s + 192.·4-s − 255.·5-s − 1.07e3·7-s + 1.15e3·8-s − 4.57e3·10-s − 4.74e3·11-s + 5.72e3·13-s − 1.92e4·14-s − 3.99e3·16-s + 1.28e4·17-s − 5.21e4·19-s − 4.91e4·20-s − 8.48e4·22-s + 9.82e3·23-s − 1.29e4·25-s + 1.02e5·26-s − 2.06e5·28-s + 1.96e5·29-s + 2.72e5·31-s − 2.19e5·32-s + 2.30e5·34-s + 2.73e5·35-s + 3.09e5·37-s − 9.32e5·38-s − 2.94e5·40-s + 4.21e5·41-s + ⋯
L(s)  = 1  + 1.58·2-s + 1.50·4-s − 0.913·5-s − 1.18·7-s + 0.796·8-s − 1.44·10-s − 1.07·11-s + 0.722·13-s − 1.87·14-s − 0.243·16-s + 0.634·17-s − 1.74·19-s − 1.37·20-s − 1.69·22-s + 0.168·23-s − 0.165·25-s + 1.14·26-s − 1.77·28-s + 1.49·29-s + 1.64·31-s − 1.18·32-s + 1.00·34-s + 1.08·35-s + 1.00·37-s − 2.75·38-s − 0.727·40-s + 0.954·41-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

\(L(4)\) \(\approx\) \(2.942700806\)
\(L(\frac12)\) \(\approx\) \(2.942700806\)
\(L(\frac{9}{2})\) 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 \( 1 - 17.8T + 128T^{2} \)
5 \( 1 + 255.T + 7.81e4T^{2} \)
7 \( 1 + 1.07e3T + 8.23e5T^{2} \)
11 \( 1 + 4.74e3T + 1.94e7T^{2} \)
13 \( 1 - 5.72e3T + 6.27e7T^{2} \)
17 \( 1 - 1.28e4T + 4.10e8T^{2} \)
19 \( 1 + 5.21e4T + 8.93e8T^{2} \)
23 \( 1 - 9.82e3T + 3.40e9T^{2} \)
29 \( 1 - 1.96e5T + 1.72e10T^{2} \)
31 \( 1 - 2.72e5T + 2.75e10T^{2} \)
37 \( 1 - 3.09e5T + 9.49e10T^{2} \)
41 \( 1 - 4.21e5T + 1.94e11T^{2} \)
43 \( 1 + 4.06e5T + 2.71e11T^{2} \)
47 \( 1 - 2.61e5T + 5.06e11T^{2} \)
53 \( 1 + 7.38e5T + 1.17e12T^{2} \)
61 \( 1 - 2.33e5T + 3.14e12T^{2} \)
67 \( 1 - 3.54e6T + 6.06e12T^{2} \)
71 \( 1 + 1.03e6T + 9.09e12T^{2} \)
73 \( 1 + 5.88e6T + 1.10e13T^{2} \)
79 \( 1 - 4.34e6T + 1.92e13T^{2} \)
83 \( 1 - 4.12e6T + 2.71e13T^{2} \)
89 \( 1 - 7.32e6T + 4.42e13T^{2} \)
97 \( 1 - 1.46e7T + 8.07e13T^{2} \)
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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

−9.997635993228845144294837008739, −8.602068908460455139915952687442, −7.74643439625549229723709433230, −6.49775850126206883264624196917, −6.11013617373486887483683007649, −4.83256692298377482963196025428, −4.07017995241208080367417322300, −3.20538350208371886063168761766, −2.48321009316638548293329689749, −0.55254259129790807462441121762, 0.55254259129790807462441121762, 2.48321009316638548293329689749, 3.20538350208371886063168761766, 4.07017995241208080367417322300, 4.83256692298377482963196025428, 6.11013617373486887483683007649, 6.49775850126206883264624196917, 7.74643439625549229723709433230, 8.602068908460455139915952687442, 9.997635993228845144294837008739

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