Properties

Degree $2$
Conductor $280$
Sign $-1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 5-s − 7-s − 2·9-s − 5·11-s + 13-s + 15-s + 3·17-s − 6·19-s + 21-s − 6·23-s + 25-s + 5·27-s − 9·29-s + 5·33-s + 35-s + 6·37-s − 39-s + 8·41-s + 6·43-s + 2·45-s + 3·47-s + 49-s − 3·51-s − 12·53-s + 5·55-s + 6·57-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.447·5-s − 0.377·7-s − 2/3·9-s − 1.50·11-s + 0.277·13-s + 0.258·15-s + 0.727·17-s − 1.37·19-s + 0.218·21-s − 1.25·23-s + 1/5·25-s + 0.962·27-s − 1.67·29-s + 0.870·33-s + 0.169·35-s + 0.986·37-s − 0.160·39-s + 1.24·41-s + 0.914·43-s + 0.298·45-s + 0.437·47-s + 1/7·49-s − 0.420·51-s − 1.64·53-s + 0.674·55-s + 0.794·57-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-1$
Motivic weight: \(1\)
Character: $\chi_{280} (1, \cdot )$
Sato-Tate group: $\mathrm{SU}(2)$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + T \)
7 \( 1 + T \)
good3 \( 1 + T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 - T + p T^{2} \)
17 \( 1 - 3 T + p T^{2} \)
19 \( 1 + 6 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 + 9 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 - 6 T + p T^{2} \)
41 \( 1 - 8 T + p T^{2} \)
43 \( 1 - 6 T + p T^{2} \)
47 \( 1 - 3 T + p T^{2} \)
53 \( 1 + 12 T + p T^{2} \)
59 \( 1 - 8 T + p T^{2} \)
61 \( 1 + 4 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 - 8 T + p T^{2} \)
73 \( 1 - 10 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 12 T + p T^{2} \)
89 \( 1 + 16 T + p T^{2} \)
97 \( 1 - 7 T + p T^{2} \)
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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

−19.85128827696897, −18.92528904075433, −18.30580092943846, −17.38388198652008, −16.52857464293880, −15.93356258769073, −15.02173193514428, −14.09700676512850, −12.92925201121896, −12.41756776308512, −11.20832939807863, −10.71481393224656, −9.607085486774063, −8.321037332106297, −7.610688958525628, −6.171063151213146, −5.457344375750090, −4.061826738127859, −2.604676005557562, 0, 2.604676005557562, 4.061826738127859, 5.457344375750090, 6.171063151213146, 7.610688958525628, 8.321037332106297, 9.607085486774063, 10.71481393224656, 11.20832939807863, 12.41756776308512, 12.92925201121896, 14.09700676512850, 15.02173193514428, 15.93356258769073, 16.52857464293880, 17.38388198652008, 18.30580092943846, 18.92528904075433, 19.85128827696897

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