Properties

Degree $2$
Conductor $287$
Sign $1$
Motivic weight $0$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.445·2-s + 1.80·3-s − 0.801·4-s − 0.801·6-s − 7-s + 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s + 0.445·14-s + 0.445·16-s − 1.24·17-s − 18-s − 1.24·19-s − 1.80·21-s − 1.80·23-s + 1.44·24-s + 25-s − 0.198·26-s + 2.24·27-s + 0.801·28-s − 32-s + 0.554·34-s − 1.80·36-s − 0.445·37-s + 0.554·38-s + 0.801·39-s + ⋯
L(s)  = 1  − 0.445·2-s + 1.80·3-s − 0.801·4-s − 0.801·6-s − 7-s + 0.801·8-s + 2.24·9-s − 1.44·12-s + 0.445·13-s + 0.445·14-s + 0.445·16-s − 1.24·17-s − 18-s − 1.24·19-s − 1.80·21-s − 1.80·23-s + 1.44·24-s + 25-s − 0.198·26-s + 2.24·27-s + 0.801·28-s − 32-s + 0.554·34-s − 1.80·36-s − 0.445·37-s + 0.554·38-s + 0.801·39-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(287\)    =    \(7 \cdot 41\)
Sign: $1$
Motivic weight: \(0\)
Character: $\chi_{287} (286, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 287,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8331120063\)
\(L(\frac12)\) \(\approx\) \(0.8331120063\)
\(L(1)\) 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 \( 1 + 0.445T + T^{2} \)
3 \( 1 - 1.80T + T^{2} \)
5 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 - 0.445T + T^{2} \)
17 \( 1 + 1.24T + T^{2} \)
19 \( 1 + 1.24T + T^{2} \)
23 \( 1 + 1.80T + T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + 0.445T + T^{2} \)
43 \( 1 - 1.24T + T^{2} \)
47 \( 1 - 0.445T + T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - 0.445T + T^{2} \)
97 \( 1 - 1.80T + T^{2} \)
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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

−12.50931839944755703134568162202, −10.57029978364407735167710001294, −9.893891954502005210537864105492, −8.870986271392845686947241209055, −8.667662838320876772337889413528, −7.56704313749342360090975818805, −6.40954737609072778002823296601, −4.39606637296033471643904162341, −3.60680270082806898606986348745, −2.19049510164046585083139919070, 2.19049510164046585083139919070, 3.60680270082806898606986348745, 4.39606637296033471643904162341, 6.40954737609072778002823296601, 7.56704313749342360090975818805, 8.667662838320876772337889413528, 8.870986271392845686947241209055, 9.893891954502005210537864105492, 10.57029978364407735167710001294, 12.50931839944755703134568162202

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