Properties

Label 2-280-1.1-c1-0-4
Degree $2$
Conductor $280$
Sign $-1$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3·3-s + 5-s + 7-s + 6·9-s − 5·11-s − 5·13-s − 3·15-s − 7·17-s − 2·19-s − 3·21-s − 2·23-s + 25-s − 9·27-s + 7·29-s + 4·31-s + 15·33-s + 35-s − 6·37-s + 15·39-s − 12·41-s − 2·43-s + 6·45-s + 47-s + 49-s + 21·51-s − 5·55-s + 6·57-s + ⋯
L(s)  = 1  − 1.73·3-s + 0.447·5-s + 0.377·7-s + 2·9-s − 1.50·11-s − 1.38·13-s − 0.774·15-s − 1.69·17-s − 0.458·19-s − 0.654·21-s − 0.417·23-s + 1/5·25-s − 1.73·27-s + 1.29·29-s + 0.718·31-s + 2.61·33-s + 0.169·35-s − 0.986·37-s + 2.40·39-s − 1.87·41-s − 0.304·43-s + 0.894·45-s + 0.145·47-s + 1/7·49-s + 2.94·51-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: $-1$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: yes
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
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 \( 1 + p T + p T^{2} \)
11 \( 1 + 5 T + p T^{2} \)
13 \( 1 + 5 T + p T^{2} \)
17 \( 1 + 7 T + p T^{2} \)
19 \( 1 + 2 T + p T^{2} \)
23 \( 1 + 2 T + p T^{2} \)
29 \( 1 - 7 T + p T^{2} \)
31 \( 1 - 4 T + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 12 T + p T^{2} \)
43 \( 1 + 2 T + p T^{2} \)
47 \( 1 - T + p T^{2} \)
53 \( 1 + p T^{2} \)
59 \( 1 + 4 T + p T^{2} \)
61 \( 1 - 4 T + p T^{2} \)
67 \( 1 - 8 T + p T^{2} \)
71 \( 1 + p T^{2} \)
73 \( 1 - 6 T + p T^{2} \)
79 \( 1 + 3 T + p T^{2} \)
83 \( 1 + 4 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 13 T + p T^{2} \)
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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

−11.37548530993775479236655988699, −10.43970270366396486648555968518, −10.04608158910457177964610000230, −8.437696063431547440013268278023, −7.13289292005793470052045653622, −6.32900819422385960555488032719, −5.09619695718514137207067111766, −4.73560872492356455619359024880, −2.25342723502104844991389935790, 0, 2.25342723502104844991389935790, 4.73560872492356455619359024880, 5.09619695718514137207067111766, 6.32900819422385960555488032719, 7.13289292005793470052045653622, 8.437696063431547440013268278023, 10.04608158910457177964610000230, 10.43970270366396486648555968518, 11.37548530993775479236655988699

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