Properties

Label 2-280-1.1-c3-0-9
Degree $2$
Conductor $280$
Sign $1$
Analytic cond. $16.5205$
Root an. cond. $4.06454$
Motivic weight $3$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7·3-s + 5·5-s + 7·7-s + 22·9-s + 9·11-s + 23·13-s + 35·15-s + 41·17-s + 34·19-s + 49·21-s − 6·23-s + 25·25-s − 35·27-s + 131·29-s + 4·31-s + 63·33-s + 35·35-s + 26·37-s + 161·39-s − 260·41-s − 190·43-s + 110·45-s + 167·47-s + 49·49-s + 287·51-s − 368·53-s + 45·55-s + ⋯
L(s)  = 1  + 1.34·3-s + 0.447·5-s + 0.377·7-s + 0.814·9-s + 0.246·11-s + 0.490·13-s + 0.602·15-s + 0.584·17-s + 0.410·19-s + 0.509·21-s − 0.0543·23-s + 1/5·25-s − 0.249·27-s + 0.838·29-s + 0.0231·31-s + 0.332·33-s + 0.169·35-s + 0.115·37-s + 0.661·39-s − 0.990·41-s − 0.673·43-s + 0.364·45-s + 0.518·47-s + 1/7·49-s + 0.788·51-s − 0.953·53-s + 0.110·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $1$
Analytic conductor: \(16.5205\)
Root analytic conductor: \(4.06454\)
Motivic weight: \(3\)
Rational: yes
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :3/2),\ 1)\)

Particular Values

\(L(2)\) \(\approx\) \(3.331440624\)
\(L(\frac12)\) \(\approx\) \(3.331440624\)
\(L(\frac{5}{2})\) 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 \( 1 - p T \)
7 \( 1 - p T \)
good3 \( 1 - 7 T + p^{3} T^{2} \)
11 \( 1 - 9 T + p^{3} T^{2} \)
13 \( 1 - 23 T + p^{3} T^{2} \)
17 \( 1 - 41 T + p^{3} T^{2} \)
19 \( 1 - 34 T + p^{3} T^{2} \)
23 \( 1 + 6 T + p^{3} T^{2} \)
29 \( 1 - 131 T + p^{3} T^{2} \)
31 \( 1 - 4 T + p^{3} T^{2} \)
37 \( 1 - 26 T + p^{3} T^{2} \)
41 \( 1 + 260 T + p^{3} T^{2} \)
43 \( 1 + 190 T + p^{3} T^{2} \)
47 \( 1 - 167 T + p^{3} T^{2} \)
53 \( 1 + 368 T + p^{3} T^{2} \)
59 \( 1 - 324 T + p^{3} T^{2} \)
61 \( 1 + 164 T + p^{3} T^{2} \)
67 \( 1 - 200 T + p^{3} T^{2} \)
71 \( 1 - 784 T + p^{3} T^{2} \)
73 \( 1 + 410 T + p^{3} T^{2} \)
79 \( 1 - 1211 T + p^{3} T^{2} \)
83 \( 1 + 1132 T + p^{3} T^{2} \)
89 \( 1 + 72 T + p^{3} T^{2} \)
97 \( 1 + 707 T + p^{3} T^{2} \)
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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

−11.41659600191462674991464103431, −10.23110738110269179470316324748, −9.398810084479949208641856549514, −8.527137718400344568353115230964, −7.82021407448414437659586636235, −6.59890538987355797608406616663, −5.26184375719422797650678122604, −3.83367019491121801595879458000, −2.75695599343501972994946863841, −1.46013290372806446505248008030, 1.46013290372806446505248008030, 2.75695599343501972994946863841, 3.83367019491121801595879458000, 5.26184375719422797650678122604, 6.59890538987355797608406616663, 7.82021407448414437659586636235, 8.527137718400344568353115230964, 9.398810084479949208641856549514, 10.23110738110269179470316324748, 11.41659600191462674991464103431

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