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Properties

Label 2-280-280.139-c1-0-16
Degree $2$
Conductor $280$
Sign $0.581 - 0.813i$
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  + (1.27 + 0.603i)2-s − 1.64·3-s + (1.27 + 1.54i)4-s + (1.78 − 1.34i)5-s + (−2.10 − 0.995i)6-s + (0.157 + 2.64i)7-s + (0.693 + 2.74i)8-s − 0.279·9-s + (3.09 − 0.635i)10-s + 4.28·11-s + (−2.09 − 2.54i)12-s − 0.653i·13-s + (−1.39 + 3.47i)14-s + (−2.95 + 2.21i)15-s + (−0.768 + 3.92i)16-s + 2.74·17-s + ⋯
L(s)  = 1  + (0.904 + 0.426i)2-s − 0.952·3-s + (0.635 + 0.772i)4-s + (0.800 − 0.599i)5-s + (−0.861 − 0.406i)6-s + (0.0594 + 0.998i)7-s + (0.245 + 0.969i)8-s − 0.0930·9-s + (0.979 − 0.200i)10-s + 1.29·11-s + (−0.605 − 0.735i)12-s − 0.181i·13-s + (−0.372 + 0.928i)14-s + (−0.761 + 0.571i)15-s + (−0.192 + 0.981i)16-s + 0.666·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.581 - 0.813i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.581 - 0.813i$
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: \((2,\ 280,\ (\ :1/2),\ 0.581 - 0.813i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61892 + 0.833093i\)
\(L(\frac12)\) \(\approx\) \(1.61892 + 0.833093i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.27 - 0.603i)T \)
5 \( 1 + (-1.78 + 1.34i)T \)
7 \( 1 + (-0.157 - 2.64i)T \)
good3 \( 1 + 1.64T + 3T^{2} \)
11 \( 1 - 4.28T + 11T^{2} \)
13 \( 1 + 0.653iT - 13T^{2} \)
17 \( 1 - 2.74T + 17T^{2} \)
19 \( 1 - 3.03iT - 19T^{2} \)
23 \( 1 + 7.21T + 23T^{2} \)
29 \( 1 + 7.72iT - 29T^{2} \)
31 \( 1 + 5.52T + 31T^{2} \)
37 \( 1 + 1.78T + 37T^{2} \)
41 \( 1 + 3.66iT - 41T^{2} \)
43 \( 1 + 6.86iT - 43T^{2} \)
47 \( 1 + 4.60iT - 47T^{2} \)
53 \( 1 - 7.10T + 53T^{2} \)
59 \( 1 + 12.8iT - 59T^{2} \)
61 \( 1 + 10.3T + 61T^{2} \)
67 \( 1 - 0.822iT - 67T^{2} \)
71 \( 1 - 11.4iT - 71T^{2} \)
73 \( 1 - 5.19T + 73T^{2} \)
79 \( 1 + 1.45iT - 79T^{2} \)
83 \( 1 + 4.62T + 83T^{2} \)
89 \( 1 - 14.0iT - 89T^{2} \)
97 \( 1 - 13.5T + 97T^{2} \)
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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

−12.03690466287110319161291094984, −11.62429045682090249220445599319, −10.23888491344411482070418160912, −9.055656301626746588993039502308, −8.082647431218887431189715250427, −6.49003805417382404068015422572, −5.82531127479294784123868062250, −5.28802864440412816825630304377, −3.88894384766183176582790670243, −2.04387595608509398776384828000, 1.45093228003294015141482168050, 3.28109331483856136670276658966, 4.51223372536649260183124201805, 5.72159496255858884836354615382, 6.44788251347890785815353761946, 7.25075418105337849201663338742, 9.283090562591206996969114093065, 10.27260328722300308650146449169, 10.92912873995281020954708005094, 11.65906112114094303369122563824

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