Properties

Label 2-531-1.1-c7-0-59
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  − 22.0·2-s + 357.·4-s + 492.·5-s − 1.20e3·7-s − 5.04e3·8-s − 1.08e4·10-s + 7.75e3·11-s + 1.92e3·13-s + 2.66e4·14-s + 6.55e4·16-s + 2.18e4·17-s − 3.16e4·19-s + 1.75e5·20-s − 1.70e5·22-s − 7.60e4·23-s + 1.64e5·25-s − 4.24e4·26-s − 4.32e5·28-s + 1.07e5·29-s − 5.68e4·31-s − 7.96e5·32-s − 4.81e5·34-s − 5.95e5·35-s − 7.25e4·37-s + 6.97e5·38-s − 2.48e6·40-s + 5.98e5·41-s + ⋯
L(s)  = 1  − 1.94·2-s + 2.79·4-s + 1.76·5-s − 1.33·7-s − 3.48·8-s − 3.43·10-s + 1.75·11-s + 0.243·13-s + 2.59·14-s + 3.99·16-s + 1.07·17-s − 1.05·19-s + 4.91·20-s − 3.42·22-s − 1.30·23-s + 2.10·25-s − 0.473·26-s − 3.72·28-s + 0.818·29-s − 0.342·31-s − 4.29·32-s − 2.10·34-s − 2.34·35-s − 0.235·37-s + 2.06·38-s − 6.14·40-s + 1.35·41-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

\(L(4)\) \(\approx\) \(1.315562953\)
\(L(\frac12)\) \(\approx\) \(1.315562953\)
\(L(\frac{9}{2})\) 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 \( 1 + 22.0T + 128T^{2} \)
5 \( 1 - 492.T + 7.81e4T^{2} \)
7 \( 1 + 1.20e3T + 8.23e5T^{2} \)
11 \( 1 - 7.75e3T + 1.94e7T^{2} \)
13 \( 1 - 1.92e3T + 6.27e7T^{2} \)
17 \( 1 - 2.18e4T + 4.10e8T^{2} \)
19 \( 1 + 3.16e4T + 8.93e8T^{2} \)
23 \( 1 + 7.60e4T + 3.40e9T^{2} \)
29 \( 1 - 1.07e5T + 1.72e10T^{2} \)
31 \( 1 + 5.68e4T + 2.75e10T^{2} \)
37 \( 1 + 7.25e4T + 9.49e10T^{2} \)
41 \( 1 - 5.98e5T + 1.94e11T^{2} \)
43 \( 1 + 1.27e5T + 2.71e11T^{2} \)
47 \( 1 + 2.07e4T + 5.06e11T^{2} \)
53 \( 1 + 1.52e6T + 1.17e12T^{2} \)
61 \( 1 - 2.40e6T + 3.14e12T^{2} \)
67 \( 1 + 6.06e5T + 6.06e12T^{2} \)
71 \( 1 - 3.52e6T + 9.09e12T^{2} \)
73 \( 1 - 1.27e6T + 1.10e13T^{2} \)
79 \( 1 - 3.89e6T + 1.92e13T^{2} \)
83 \( 1 - 4.03e6T + 2.71e13T^{2} \)
89 \( 1 + 5.97e6T + 4.42e13T^{2} \)
97 \( 1 + 1.98e6T + 8.07e13T^{2} \)
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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

−9.635489909198624376430261363587, −9.177688963085031692776093994733, −8.279613488240597421024574823626, −6.88266860677894354814170131176, −6.30931492252254435466979775148, −5.92547062613901041023172160369, −3.51734910302968189781997952255, −2.37253458467923084436734708680, −1.55946600107316248983192049630, −0.69670242489744105333993569248, 0.69670242489744105333993569248, 1.55946600107316248983192049630, 2.37253458467923084436734708680, 3.51734910302968189781997952255, 5.92547062613901041023172160369, 6.30931492252254435466979775148, 6.88266860677894354814170131176, 8.279613488240597421024574823626, 9.177688963085031692776093994733, 9.635489909198624376430261363587

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