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Properties

Label 2-280-35.27-c1-0-5
Degree $2$
Conductor $280$
Sign $0.984 - 0.173i$
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  + (0.730 + 0.730i)3-s + (1.51 − 1.64i)5-s + (−1.08 + 2.41i)7-s − 1.93i·9-s + 5.95·11-s + (0.921 + 0.921i)13-s + (2.30 − 0.0988i)15-s + (−2.02 + 2.02i)17-s − 3.29·19-s + (−2.55 + 0.970i)21-s + (0.0544 − 0.0544i)23-s + (−0.427 − 4.98i)25-s + (3.60 − 3.60i)27-s + 3.78i·29-s + 4.88i·31-s + ⋯
L(s)  = 1  + (0.421 + 0.421i)3-s + (0.676 − 0.736i)5-s + (−0.409 + 0.912i)7-s − 0.644i·9-s + 1.79·11-s + (0.255 + 0.255i)13-s + (0.595 − 0.0255i)15-s + (−0.491 + 0.491i)17-s − 0.755·19-s + (−0.557 + 0.211i)21-s + (0.0113 − 0.0113i)23-s + (−0.0855 − 0.996i)25-s + (0.693 − 0.693i)27-s + 0.703i·29-s + 0.876i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.984 - 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.984 - 0.173i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.984 - 0.173i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.61655 + 0.141049i\)
\(L(\frac12)\) \(\approx\) \(1.61655 + 0.141049i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
5 \( 1 + (-1.51 + 1.64i)T \)
7 \( 1 + (1.08 - 2.41i)T \)
good3 \( 1 + (-0.730 - 0.730i)T + 3iT^{2} \)
11 \( 1 - 5.95T + 11T^{2} \)
13 \( 1 + (-0.921 - 0.921i)T + 13iT^{2} \)
17 \( 1 + (2.02 - 2.02i)T - 17iT^{2} \)
19 \( 1 + 3.29T + 19T^{2} \)
23 \( 1 + (-0.0544 + 0.0544i)T - 23iT^{2} \)
29 \( 1 - 3.78iT - 29T^{2} \)
31 \( 1 - 4.88iT - 31T^{2} \)
37 \( 1 + (3.20 + 3.20i)T + 37iT^{2} \)
41 \( 1 + 10.6iT - 41T^{2} \)
43 \( 1 + (1.60 - 1.60i)T - 43iT^{2} \)
47 \( 1 + (2.64 - 2.64i)T - 47iT^{2} \)
53 \( 1 + (9.16 - 9.16i)T - 53iT^{2} \)
59 \( 1 + 13.3T + 59T^{2} \)
61 \( 1 + 11.2iT - 61T^{2} \)
67 \( 1 + (6.43 + 6.43i)T + 67iT^{2} \)
71 \( 1 + 8.51T + 71T^{2} \)
73 \( 1 + (-8.66 - 8.66i)T + 73iT^{2} \)
79 \( 1 - 1.76iT - 79T^{2} \)
83 \( 1 + (-2.36 - 2.36i)T + 83iT^{2} \)
89 \( 1 - 9.73T + 89T^{2} \)
97 \( 1 + (-2.05 + 2.05i)T - 97iT^{2} \)
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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

−12.32647783877192134029116470313, −10.88937404681603120083197499721, −9.601441644088658741941938977484, −9.009936361072756433258979786682, −8.642350627450075330621125451803, −6.62284395227776602760162444750, −6.03919006647763269737902091521, −4.56819785884333727976953142420, −3.45520885765028924846133284143, −1.73220906351565905288897793293, 1.69509133945187316702662114854, 3.16507628765166322236177226777, 4.44734046746540886739079112575, 6.24010316078879585380917040539, 6.80263194295418073627726478493, 7.83548120132940873466707372967, 9.070073416056081216380300827833, 9.940651490149510382380037019667, 10.85582788438399549662801573000, 11.70916280769774459528435858565

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