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Properties

Label 2-280-280.139-c1-0-21
Degree $2$
Conductor $280$
Sign $0.845 - 0.534i$
Analytic cond. $2.23581$
Root an. cond. $1.49526$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s + 2.00·4-s + 2.23i·5-s + (1.41 + 2.23i)7-s + 2.82·8-s − 3·9-s + 3.16i·10-s − 2·11-s − 4.47i·13-s + (2.00 + 3.16i)14-s + 4.00·16-s − 4.24·18-s − 6.32i·19-s + 4.47i·20-s − 2.82·22-s + 8.48·23-s + ⋯
L(s)  = 1  + 1.00·2-s + 1.00·4-s + 0.999i·5-s + (0.534 + 0.845i)7-s + 1.00·8-s − 9-s + 1.00i·10-s − 0.603·11-s − 1.24i·13-s + (0.534 + 0.845i)14-s + 1.00·16-s − 1.00·18-s − 1.45i·19-s + 1.00i·20-s − 0.603·22-s + 1.76·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.845 - 0.534i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.845 - 0.534i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (139, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.845 - 0.534i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17563 + 0.630260i\)
\(L(\frac12)\) \(\approx\) \(2.17563 + 0.630260i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 - 2.23iT \)
7 \( 1 + (-1.41 - 2.23i)T \)
good3 \( 1 + 3T^{2} \)
11 \( 1 + 2T + 11T^{2} \)
13 \( 1 + 4.47iT - 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 + 6.32iT - 19T^{2} \)
23 \( 1 - 8.48T + 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 11.3T + 37T^{2} \)
41 \( 1 + 12.6iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 13.4iT - 47T^{2} \)
53 \( 1 + 5.65T + 53T^{2} \)
59 \( 1 - 6.32iT - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 12.6iT - 89T^{2} \)
97 \( 1 + 97T^{2} \)
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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

−11.95298048073052054700082457652, −10.94801309475838992351736972824, −10.71757440336990693655142013303, −8.991470529921574896676335751041, −7.84575001163635551796609561235, −6.87066805789182194305403868038, −5.66719393853226538374610995631, −5.05661512192263075397131105126, −3.14624889166483501802787244592, −2.56121618158740582142065350812, 1.67728263803887682959896456342, 3.48059889040351849661163209822, 4.68321640491882346727254862445, 5.39724669473264633519030471226, 6.68437098038172920798169694916, 7.83456578543562229550049433579, 8.718779477908862890402083894290, 10.11299833159821396954814596655, 11.18384574350707822592441272191, 11.81580721539414081594820189781

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