Properties

Label 2-287-1.1-c1-0-11
Degree $2$
Conductor $287$
Sign $1$
Analytic cond. $2.29170$
Root an. cond. $1.51383$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2.46·2-s − 2.61·3-s + 4.06·4-s + 2.85·5-s − 6.42·6-s − 7-s + 5.07·8-s + 3.81·9-s + 7.03·10-s + 3.40·11-s − 10.5·12-s − 2.37·13-s − 2.46·14-s − 7.45·15-s + 4.36·16-s + 5.67·17-s + 9.39·18-s − 7.43·19-s + 11.5·20-s + 2.61·21-s + 8.37·22-s + 1.38·23-s − 13.2·24-s + 3.15·25-s − 5.83·26-s − 2.12·27-s − 4.06·28-s + ⋯
L(s)  = 1  + 1.74·2-s − 1.50·3-s + 2.03·4-s + 1.27·5-s − 2.62·6-s − 0.377·7-s + 1.79·8-s + 1.27·9-s + 2.22·10-s + 1.02·11-s − 3.05·12-s − 0.657·13-s − 0.657·14-s − 1.92·15-s + 1.09·16-s + 1.37·17-s + 2.21·18-s − 1.70·19-s + 2.59·20-s + 0.569·21-s + 1.78·22-s + 0.289·23-s − 2.70·24-s + 0.631·25-s − 1.14·26-s − 0.409·27-s − 0.767·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 287 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $1$
Analytic conductor: \(2.29170\)
Root analytic conductor: \(1.51383\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.482443985\)
\(L(\frac12)\) \(\approx\) \(2.482443985\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad7 \( 1 + T \)
41 \( 1 - T \)
good2 \( 1 - 2.46T + 2T^{2} \)
3 \( 1 + 2.61T + 3T^{2} \)
5 \( 1 - 2.85T + 5T^{2} \)
11 \( 1 - 3.40T + 11T^{2} \)
13 \( 1 + 2.37T + 13T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + 7.43T + 19T^{2} \)
23 \( 1 - 1.38T + 23T^{2} \)
29 \( 1 + 5.08T + 29T^{2} \)
31 \( 1 + 10.4T + 31T^{2} \)
37 \( 1 + 0.480T + 37T^{2} \)
43 \( 1 + 4.77T + 43T^{2} \)
47 \( 1 - 1.26T + 47T^{2} \)
53 \( 1 + 2.05T + 53T^{2} \)
59 \( 1 - 13.8T + 59T^{2} \)
61 \( 1 + 8.73T + 61T^{2} \)
67 \( 1 - 3.63T + 67T^{2} \)
71 \( 1 + 10.2T + 71T^{2} \)
73 \( 1 - 16.3T + 73T^{2} \)
79 \( 1 + 1.82T + 79T^{2} \)
83 \( 1 - 16.3T + 83T^{2} \)
89 \( 1 - 9.48T + 89T^{2} \)
97 \( 1 - 2.62T + 97T^{2} \)
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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

−12.14459614533544970304792443173, −11.17960657744998178045041421755, −10.38942025788974507424243508190, −9.354530744180046730522916629988, −7.09450313097692349860353663739, −6.29906134716818663296225780485, −5.72514990553029491632028922693, −4.99084112716440111855460068245, −3.72523009581308654023372077230, −1.94750764742205736906413678243, 1.94750764742205736906413678243, 3.72523009581308654023372077230, 4.99084112716440111855460068245, 5.72514990553029491632028922693, 6.29906134716818663296225780485, 7.09450313097692349860353663739, 9.354530744180046730522916629988, 10.38942025788974507424243508190, 11.17960657744998178045041421755, 12.14459614533544970304792443173

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