Properties

Label 2-280-1.1-c5-0-19
Degree $2$
Conductor $280$
Sign $-1$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 17.3·3-s + 25·5-s + 49·7-s + 58.9·9-s + 12.2·11-s − 678.·13-s − 434.·15-s + 1.50e3·17-s − 1.39e3·19-s − 851.·21-s + 4.19e3·23-s + 625·25-s + 3.19e3·27-s + 6.73e3·29-s − 9.61e3·31-s − 213.·33-s + 1.22e3·35-s − 4.55e3·37-s + 1.17e4·39-s + 1.82e4·41-s − 1.84e4·43-s + 1.47e3·45-s − 2.24e4·47-s + 2.40e3·49-s − 2.62e4·51-s − 2.90e4·53-s + 307.·55-s + ⋯
L(s)  = 1  − 1.11·3-s + 0.447·5-s + 0.377·7-s + 0.242·9-s + 0.0306·11-s − 1.11·13-s − 0.498·15-s + 1.26·17-s − 0.886·19-s − 0.421·21-s + 1.65·23-s + 0.200·25-s + 0.844·27-s + 1.48·29-s − 1.79·31-s − 0.0341·33-s + 0.169·35-s − 0.546·37-s + 1.24·39-s + 1.69·41-s − 1.52·43-s + 0.108·45-s − 1.48·47-s + 0.142·49-s − 1.41·51-s − 1.42·53-s + 0.0137·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-1$
Analytic conductor: \(44.9074\)
Root analytic conductor: \(6.70130\)
Motivic weight: \(5\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 280,\ (\ :5/2),\ -1)\)

Particular Values

\(L(3)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{7}{2})\) 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 \( 1 - 25T \)
7 \( 1 - 49T \)
good3 \( 1 + 17.3T + 243T^{2} \)
11 \( 1 - 12.2T + 1.61e5T^{2} \)
13 \( 1 + 678.T + 3.71e5T^{2} \)
17 \( 1 - 1.50e3T + 1.41e6T^{2} \)
19 \( 1 + 1.39e3T + 2.47e6T^{2} \)
23 \( 1 - 4.19e3T + 6.43e6T^{2} \)
29 \( 1 - 6.73e3T + 2.05e7T^{2} \)
31 \( 1 + 9.61e3T + 2.86e7T^{2} \)
37 \( 1 + 4.55e3T + 6.93e7T^{2} \)
41 \( 1 - 1.82e4T + 1.15e8T^{2} \)
43 \( 1 + 1.84e4T + 1.47e8T^{2} \)
47 \( 1 + 2.24e4T + 2.29e8T^{2} \)
53 \( 1 + 2.90e4T + 4.18e8T^{2} \)
59 \( 1 + 6.52e3T + 7.14e8T^{2} \)
61 \( 1 - 8.73e3T + 8.44e8T^{2} \)
67 \( 1 + 1.58e4T + 1.35e9T^{2} \)
71 \( 1 + 2.19e4T + 1.80e9T^{2} \)
73 \( 1 + 2.97e4T + 2.07e9T^{2} \)
79 \( 1 + 6.52e4T + 3.07e9T^{2} \)
83 \( 1 + 1.02e5T + 3.93e9T^{2} \)
89 \( 1 - 1.16e5T + 5.58e9T^{2} \)
97 \( 1 + 1.18e5T + 8.58e9T^{2} \)
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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

−10.65363068317777052560586901972, −9.809792501741617540934767375212, −8.669132243321898517211637360329, −7.40812181585284365082860264846, −6.43155283216311769250923835560, −5.37817334688003719224045933239, −4.74625704202239531172674058320, −2.94438265540481431091055211129, −1.37505942126249257328326042987, 0, 1.37505942126249257328326042987, 2.94438265540481431091055211129, 4.74625704202239531172674058320, 5.37817334688003719224045933239, 6.43155283216311769250923835560, 7.40812181585284365082860264846, 8.669132243321898517211637360329, 9.809792501741617540934767375212, 10.65363068317777052560586901972

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