Properties

Label 2-531-1.1-c7-0-142
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  + 16.2·2-s + 137.·4-s − 495.·5-s + 565.·7-s + 155.·8-s − 8.07e3·10-s + 826.·11-s + 1.14e4·13-s + 9.21e3·14-s − 1.50e4·16-s + 2.35e4·17-s − 5.35e4·19-s − 6.81e4·20-s + 1.34e4·22-s + 8.71e4·23-s + 1.67e5·25-s + 1.87e5·26-s + 7.77e4·28-s − 2.67e4·29-s − 2.31e5·31-s − 2.65e5·32-s + 3.83e5·34-s − 2.80e5·35-s + 4.09e5·37-s − 8.72e5·38-s − 7.69e4·40-s − 1.89e5·41-s + ⋯
L(s)  = 1  + 1.44·2-s + 1.07·4-s − 1.77·5-s + 0.622·7-s + 0.107·8-s − 2.55·10-s + 0.187·11-s + 1.44·13-s + 0.897·14-s − 0.919·16-s + 1.16·17-s − 1.79·19-s − 1.90·20-s + 0.269·22-s + 1.49·23-s + 2.14·25-s + 2.08·26-s + 0.669·28-s − 0.203·29-s − 1.39·31-s − 1.43·32-s + 1.67·34-s − 1.10·35-s + 1.33·37-s − 2.57·38-s − 0.190·40-s − 0.429·41-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 \( 1 - 16.2T + 128T^{2} \)
5 \( 1 + 495.T + 7.81e4T^{2} \)
7 \( 1 - 565.T + 8.23e5T^{2} \)
11 \( 1 - 826.T + 1.94e7T^{2} \)
13 \( 1 - 1.14e4T + 6.27e7T^{2} \)
17 \( 1 - 2.35e4T + 4.10e8T^{2} \)
19 \( 1 + 5.35e4T + 8.93e8T^{2} \)
23 \( 1 - 8.71e4T + 3.40e9T^{2} \)
29 \( 1 + 2.67e4T + 1.72e10T^{2} \)
31 \( 1 + 2.31e5T + 2.75e10T^{2} \)
37 \( 1 - 4.09e5T + 9.49e10T^{2} \)
41 \( 1 + 1.89e5T + 1.94e11T^{2} \)
43 \( 1 + 8.49e5T + 2.71e11T^{2} \)
47 \( 1 - 9.02e5T + 5.06e11T^{2} \)
53 \( 1 - 3.24e5T + 1.17e12T^{2} \)
61 \( 1 + 1.58e6T + 3.14e12T^{2} \)
67 \( 1 + 2.57e6T + 6.06e12T^{2} \)
71 \( 1 - 6.78e5T + 9.09e12T^{2} \)
73 \( 1 - 2.50e6T + 1.10e13T^{2} \)
79 \( 1 + 7.33e6T + 1.92e13T^{2} \)
83 \( 1 + 3.91e6T + 2.71e13T^{2} \)
89 \( 1 + 3.80e6T + 4.42e13T^{2} \)
97 \( 1 + 7.68e6T + 8.07e13T^{2} \)
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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

−8.917831105155558723983911141966, −8.275658723955946323082032013924, −7.31338330530635454493240632931, −6.37345991335947831280755270789, −5.28958016440438704523616008340, −4.32930711581838078778192399862, −3.79262522046643302789600907022, −2.99744321086221120475249224103, −1.34241343611899950859386596028, 0, 1.34241343611899950859386596028, 2.99744321086221120475249224103, 3.79262522046643302789600907022, 4.32930711581838078778192399862, 5.28958016440438704523616008340, 6.37345991335947831280755270789, 7.31338330530635454493240632931, 8.275658723955946323082032013924, 8.917831105155558723983911141966

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