Properties

Label 2-280-1.1-c5-0-23
Degree $2$
Conductor $280$
Sign $-1$
Analytic cond. $44.9074$
Root an. cond. $6.70130$
Motivic weight $5$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 4·3-s + 25·5-s − 49·7-s − 227·9-s + 124·11-s + 766·13-s + 100·15-s − 1.10e3·17-s − 764·19-s − 196·21-s + 168·23-s + 625·25-s − 1.88e3·27-s − 6.86e3·29-s − 4.09e3·31-s + 496·33-s − 1.22e3·35-s − 4.68e3·37-s + 3.06e3·39-s + 1.31e4·41-s + 1.82e4·43-s − 5.67e3·45-s − 7.10e3·47-s + 2.40e3·49-s − 4.40e3·51-s − 2.00e4·53-s + 3.10e3·55-s + ⋯
L(s)  = 1  + 0.256·3-s + 0.447·5-s − 0.377·7-s − 0.934·9-s + 0.308·11-s + 1.25·13-s + 0.114·15-s − 0.924·17-s − 0.485·19-s − 0.0969·21-s + 0.0662·23-s + 1/5·25-s − 0.496·27-s − 1.51·29-s − 0.765·31-s + 0.0792·33-s − 0.169·35-s − 0.562·37-s + 0.322·39-s + 1.21·41-s + 1.50·43-s − 0.417·45-s − 0.469·47-s + 1/7·49-s − 0.237·51-s − 0.979·53-s + 0.138·55-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: yes
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 \( 1 - p^{2} T \)
7 \( 1 + p^{2} T \)
good3 \( 1 - 4 T + p^{5} T^{2} \)
11 \( 1 - 124 T + p^{5} T^{2} \)
13 \( 1 - 766 T + p^{5} T^{2} \)
17 \( 1 + 1102 T + p^{5} T^{2} \)
19 \( 1 + 764 T + p^{5} T^{2} \)
23 \( 1 - 168 T + p^{5} T^{2} \)
29 \( 1 + 6866 T + p^{5} T^{2} \)
31 \( 1 + 4096 T + p^{5} T^{2} \)
37 \( 1 + 4682 T + p^{5} T^{2} \)
41 \( 1 - 13130 T + p^{5} T^{2} \)
43 \( 1 - 18220 T + p^{5} T^{2} \)
47 \( 1 + 7104 T + p^{5} T^{2} \)
53 \( 1 + 20026 T + p^{5} T^{2} \)
59 \( 1 + 38964 T + p^{5} T^{2} \)
61 \( 1 + 56274 T + p^{5} T^{2} \)
67 \( 1 + 24060 T + p^{5} T^{2} \)
71 \( 1 + 31896 T + p^{5} T^{2} \)
73 \( 1 + 23670 T + p^{5} T^{2} \)
79 \( 1 - 37744 T + p^{5} T^{2} \)
83 \( 1 + 68204 T + p^{5} T^{2} \)
89 \( 1 + 19078 T + p^{5} T^{2} \)
97 \( 1 + 115646 T + p^{5} T^{2} \)
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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

−10.76683165766082367173587364490, −9.224262926035852895831946015060, −8.931585227828103694472737066306, −7.67287939102907538110820938844, −6.35479167625768951636807521110, −5.71787135623926833648327528383, −4.15206788173510201102169121147, −3.00391131981547898238559190865, −1.70253821510569143173827380385, 0, 1.70253821510569143173827380385, 3.00391131981547898238559190865, 4.15206788173510201102169121147, 5.71787135623926833648327528383, 6.35479167625768951636807521110, 7.67287939102907538110820938844, 8.931585227828103694472737066306, 9.224262926035852895831946015060, 10.76683165766082367173587364490

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