Properties

Label 2-531-1.1-c7-0-0
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  + 14.3·2-s + 76.6·4-s − 348.·5-s − 1.31e3·7-s − 734.·8-s − 4.99e3·10-s − 4.99e3·11-s + 3.39e3·13-s − 1.88e4·14-s − 2.03e4·16-s − 1.30e4·17-s + 8.79e3·19-s − 2.67e4·20-s − 7.14e4·22-s − 1.07e5·23-s + 4.35e4·25-s + 4.86e4·26-s − 1.00e5·28-s − 3.19e4·29-s − 1.54e5·31-s − 1.96e5·32-s − 1.87e5·34-s + 4.59e5·35-s − 5.28e5·37-s + 1.25e5·38-s + 2.56e5·40-s + 4.80e5·41-s + ⋯
L(s)  = 1  + 1.26·2-s + 0.598·4-s − 1.24·5-s − 1.45·7-s − 0.507·8-s − 1.57·10-s − 1.13·11-s + 0.428·13-s − 1.83·14-s − 1.24·16-s − 0.645·17-s + 0.294·19-s − 0.747·20-s − 1.43·22-s − 1.84·23-s + 0.558·25-s + 0.542·26-s − 0.868·28-s − 0.242·29-s − 0.930·31-s − 1.06·32-s − 0.816·34-s + 1.81·35-s − 1.71·37-s + 0.371·38-s + 0.633·40-s + 1.08·41-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: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 531,\ (\ :7/2),\ 1)\)

Particular Values

\(L(4)\) \(\approx\) \(0.008688264737\)
\(L(\frac12)\) \(\approx\) \(0.008688264737\)
\(L(\frac{9}{2})\) 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 \( 1 - 14.3T + 128T^{2} \)
5 \( 1 + 348.T + 7.81e4T^{2} \)
7 \( 1 + 1.31e3T + 8.23e5T^{2} \)
11 \( 1 + 4.99e3T + 1.94e7T^{2} \)
13 \( 1 - 3.39e3T + 6.27e7T^{2} \)
17 \( 1 + 1.30e4T + 4.10e8T^{2} \)
19 \( 1 - 8.79e3T + 8.93e8T^{2} \)
23 \( 1 + 1.07e5T + 3.40e9T^{2} \)
29 \( 1 + 3.19e4T + 1.72e10T^{2} \)
31 \( 1 + 1.54e5T + 2.75e10T^{2} \)
37 \( 1 + 5.28e5T + 9.49e10T^{2} \)
41 \( 1 - 4.80e5T + 1.94e11T^{2} \)
43 \( 1 + 4.38e5T + 2.71e11T^{2} \)
47 \( 1 + 1.23e6T + 5.06e11T^{2} \)
53 \( 1 - 1.18e6T + 1.17e12T^{2} \)
61 \( 1 - 8.73e5T + 3.14e12T^{2} \)
67 \( 1 + 8.32e5T + 6.06e12T^{2} \)
71 \( 1 + 3.15e6T + 9.09e12T^{2} \)
73 \( 1 - 2.69e6T + 1.10e13T^{2} \)
79 \( 1 - 6.93e6T + 1.92e13T^{2} \)
83 \( 1 - 3.14e6T + 2.71e13T^{2} \)
89 \( 1 - 4.70e5T + 4.42e13T^{2} \)
97 \( 1 - 1.03e7T + 8.07e13T^{2} \)
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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

−9.796810738684052584022255022955, −8.730339028938498378700093495853, −7.75621386916340681535692787075, −6.75820751841446474454338034913, −5.92360822297854178855379021256, −4.95389086878538714076948958642, −3.77082805664154464363524042057, −3.50479312642898961442672510088, −2.34083010076745466415063194339, −0.03054635719494136305975541860, 0.03054635719494136305975541860, 2.34083010076745466415063194339, 3.50479312642898961442672510088, 3.77082805664154464363524042057, 4.95389086878538714076948958642, 5.92360822297854178855379021256, 6.75820751841446474454338034913, 7.75621386916340681535692787075, 8.730339028938498378700093495853, 9.796810738684052584022255022955

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