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Properties

Label 2-280-56.27-c1-0-6
Degree $2$
Conductor $280$
Sign $0.532 - 0.846i$
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  + (−1.24 − 0.678i)2-s + 1.61i·3-s + (1.08 + 1.68i)4-s + 5-s + (1.09 − 1.99i)6-s + (2.13 + 1.56i)7-s + (−0.198 − 2.82i)8-s + 0.405·9-s + (−1.24 − 0.678i)10-s − 6.01·11-s + (−2.71 + 1.73i)12-s + 4.25·13-s + (−1.58 − 3.38i)14-s + 1.61i·15-s + (−1.66 + 3.63i)16-s + 5.42i·17-s + ⋯
L(s)  = 1  + (−0.877 − 0.479i)2-s + 0.929i·3-s + (0.540 + 0.841i)4-s + 0.447·5-s + (0.445 − 0.816i)6-s + (0.806 + 0.590i)7-s + (−0.0702 − 0.997i)8-s + 0.135·9-s + (−0.392 − 0.214i)10-s − 1.81·11-s + (−0.782 + 0.502i)12-s + 1.17·13-s + (−0.424 − 0.905i)14-s + 0.415i·15-s + (−0.416 + 0.909i)16-s + 1.31i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.532 - 0.846i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.532 - 0.846i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (251, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.532 - 0.846i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.848086 + 0.468313i\)
\(L(\frac12)\) \(\approx\) \(0.848086 + 0.468313i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.24 + 0.678i)T \)
5 \( 1 - T \)
7 \( 1 + (-2.13 - 1.56i)T \)
good3 \( 1 - 1.61iT - 3T^{2} \)
11 \( 1 + 6.01T + 11T^{2} \)
13 \( 1 - 4.25T + 13T^{2} \)
17 \( 1 - 5.42iT - 17T^{2} \)
19 \( 1 + 4.53iT - 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 - 0.376iT - 29T^{2} \)
31 \( 1 - 2.93T + 31T^{2} \)
37 \( 1 + 0.372iT - 37T^{2} \)
41 \( 1 + 5.75iT - 41T^{2} \)
43 \( 1 + 4.96T + 43T^{2} \)
47 \( 1 - 3.86T + 47T^{2} \)
53 \( 1 + 10.2iT - 53T^{2} \)
59 \( 1 - 10.5iT - 59T^{2} \)
61 \( 1 + 5.58T + 61T^{2} \)
67 \( 1 + 0.782T + 67T^{2} \)
71 \( 1 + 15.3iT - 71T^{2} \)
73 \( 1 + 8.77iT - 73T^{2} \)
79 \( 1 + 8.74iT - 79T^{2} \)
83 \( 1 + 8.42iT - 83T^{2} \)
89 \( 1 + 1.94iT - 89T^{2} \)
97 \( 1 + 3.14iT - 97T^{2} \)
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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

−11.61876391910033328384390105321, −10.62331760254483779029190155859, −10.42796274666470429339517796156, −9.179336249935187189249612433876, −8.472679622387953838550753990495, −7.52439859745685008899288386372, −5.91720289636413461743980600682, −4.76305802813230474189115975069, −3.32848696423407492408787933545, −1.88084607191702109434365472577, 1.07602007705532920140135810520, 2.42693316664145583197368751989, 4.86307616371084992503225471137, 5.98542634528804732288583988500, 7.03796878053652777314100894616, 7.88364227326854523015111067615, 8.422838795404406090924408491415, 9.927829016606157565567409861930, 10.58174731505343437904212045263, 11.47689785945099171541326384919

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