Properties

Label 2-531-1.1-c7-0-86
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7.00·2-s − 78.8·4-s − 449.·5-s − 271.·7-s − 1.44e3·8-s − 3.14e3·10-s + 3.16e3·11-s − 9.44e3·13-s − 1.90e3·14-s − 63.3·16-s + 6.79e3·17-s + 3.40e4·19-s + 3.54e4·20-s + 2.21e4·22-s + 3.12e4·23-s + 1.23e5·25-s − 6.62e4·26-s + 2.14e4·28-s − 8.27e4·29-s + 2.83e5·31-s + 1.85e5·32-s + 4.76e4·34-s + 1.22e5·35-s + 2.68e4·37-s + 2.38e5·38-s + 6.51e5·40-s + 1.95e4·41-s + ⋯
L(s)  = 1  + 0.619·2-s − 0.616·4-s − 1.60·5-s − 0.299·7-s − 1.00·8-s − 0.995·10-s + 0.716·11-s − 1.19·13-s − 0.185·14-s − 0.00386·16-s + 0.335·17-s + 1.14·19-s + 0.990·20-s + 0.444·22-s + 0.535·23-s + 1.58·25-s − 0.738·26-s + 0.184·28-s − 0.629·29-s + 1.71·31-s + 0.998·32-s + 0.207·34-s + 0.481·35-s + 0.0870·37-s + 0.706·38-s + 1.60·40-s + 0.0442·41-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: \(1\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ -1)\)

Particular Values

\(L(4)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{9}{2})\) 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 \( 1 - 7.00T + 128T^{2} \)
5 \( 1 + 449.T + 7.81e4T^{2} \)
7 \( 1 + 271.T + 8.23e5T^{2} \)
11 \( 1 - 3.16e3T + 1.94e7T^{2} \)
13 \( 1 + 9.44e3T + 6.27e7T^{2} \)
17 \( 1 - 6.79e3T + 4.10e8T^{2} \)
19 \( 1 - 3.40e4T + 8.93e8T^{2} \)
23 \( 1 - 3.12e4T + 3.40e9T^{2} \)
29 \( 1 + 8.27e4T + 1.72e10T^{2} \)
31 \( 1 - 2.83e5T + 2.75e10T^{2} \)
37 \( 1 - 2.68e4T + 9.49e10T^{2} \)
41 \( 1 - 1.95e4T + 1.94e11T^{2} \)
43 \( 1 + 4.17e5T + 2.71e11T^{2} \)
47 \( 1 - 2.76e4T + 5.06e11T^{2} \)
53 \( 1 + 2.90e4T + 1.17e12T^{2} \)
61 \( 1 - 1.53e6T + 3.14e12T^{2} \)
67 \( 1 + 2.52e6T + 6.06e12T^{2} \)
71 \( 1 - 3.35e6T + 9.09e12T^{2} \)
73 \( 1 + 1.90e6T + 1.10e13T^{2} \)
79 \( 1 - 2.26e6T + 1.92e13T^{2} \)
83 \( 1 - 5.96e6T + 2.71e13T^{2} \)
89 \( 1 + 5.11e6T + 4.42e13T^{2} \)
97 \( 1 + 1.45e5T + 8.07e13T^{2} \)
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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

−9.274309271539556676041212877930, −8.267131455545471495181568031498, −7.50050650947884286878573465574, −6.56122538244264284551418861312, −5.21302863560028495294584469692, −4.50230478090456652158841057906, −3.62568141934094170323956424917, −2.92601982474903406575168629572, −0.913253144127034297947856632617, 0, 0.913253144127034297947856632617, 2.92601982474903406575168629572, 3.62568141934094170323956424917, 4.50230478090456652158841057906, 5.21302863560028495294584469692, 6.56122538244264284551418861312, 7.50050650947884286878573465574, 8.267131455545471495181568031498, 9.274309271539556676041212877930

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