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Properties

Label 2-280-40.27-c1-0-11
Degree $2$
Conductor $280$
Sign $0.999 - 0.0151i$
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  + (−0.102 − 1.41i)2-s + (−0.977 + 0.977i)3-s + (−1.97 + 0.288i)4-s + (2.23 − 0.0665i)5-s + (1.47 + 1.27i)6-s + (−0.707 + 0.707i)7-s + (0.609 + 2.76i)8-s + 1.08i·9-s + (−0.322 − 3.14i)10-s − 0.190·11-s + (1.65 − 2.21i)12-s + (3.83 + 3.83i)13-s + (1.06 + 0.925i)14-s + (−2.12 + 2.25i)15-s + (3.83 − 1.14i)16-s + (1.66 + 1.66i)17-s + ⋯
L(s)  = 1  + (−0.0723 − 0.997i)2-s + (−0.564 + 0.564i)3-s + (−0.989 + 0.144i)4-s + (0.999 − 0.0297i)5-s + (0.603 + 0.522i)6-s + (−0.267 + 0.267i)7-s + (0.215 + 0.976i)8-s + 0.362i·9-s + (−0.102 − 0.994i)10-s − 0.0572·11-s + (0.477 − 0.640i)12-s + (1.06 + 1.06i)13-s + (0.285 + 0.247i)14-s + (−0.547 + 0.581i)15-s + (0.958 − 0.285i)16-s + (0.403 + 0.403i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0151i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0151i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.999 - 0.0151i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (267, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.999 - 0.0151i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07966 + 0.00817612i\)
\(L(\frac12)\) \(\approx\) \(1.07966 + 0.00817612i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.102 + 1.41i)T \)
5 \( 1 + (-2.23 + 0.0665i)T \)
7 \( 1 + (0.707 - 0.707i)T \)
good3 \( 1 + (0.977 - 0.977i)T - 3iT^{2} \)
11 \( 1 + 0.190T + 11T^{2} \)
13 \( 1 + (-3.83 - 3.83i)T + 13iT^{2} \)
17 \( 1 + (-1.66 - 1.66i)T + 17iT^{2} \)
19 \( 1 + 7.16iT - 19T^{2} \)
23 \( 1 + (-5.10 - 5.10i)T + 23iT^{2} \)
29 \( 1 - 0.389T + 29T^{2} \)
31 \( 1 - 5.59iT - 31T^{2} \)
37 \( 1 + (-0.807 + 0.807i)T - 37iT^{2} \)
41 \( 1 + 9.77T + 41T^{2} \)
43 \( 1 + (2.07 - 2.07i)T - 43iT^{2} \)
47 \( 1 + (2.90 - 2.90i)T - 47iT^{2} \)
53 \( 1 + (3.87 + 3.87i)T + 53iT^{2} \)
59 \( 1 + 3.53iT - 59T^{2} \)
61 \( 1 + 10.7iT - 61T^{2} \)
67 \( 1 + (-7.88 - 7.88i)T + 67iT^{2} \)
71 \( 1 + 1.44iT - 71T^{2} \)
73 \( 1 + (6.73 - 6.73i)T - 73iT^{2} \)
79 \( 1 - 9.81T + 79T^{2} \)
83 \( 1 + (0.608 - 0.608i)T - 83iT^{2} \)
89 \( 1 + 16.1iT - 89T^{2} \)
97 \( 1 + (-0.0632 - 0.0632i)T + 97iT^{2} \)
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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

−11.44277316789363100555046528759, −11.05977363225453288163834296452, −10.05723500724516162124833957882, −9.317790551819650667399468519241, −8.531077663221373352648794393687, −6.71997270512243012330722940608, −5.46107350393212927187416241610, −4.70686749165304121348506672437, −3.20244490496913238640688691679, −1.68902723561830038218364045416, 1.05092745639058469820556492246, 3.50718385043667183222949393166, 5.24464575593972498846398376074, 6.05578631649565159683292582818, 6.64613808994760510572300705261, 7.82969282067606537034505485296, 8.861785141512474986482991245790, 9.925352468049040247847427711236, 10.64152530023128509904641434063, 12.17478656595234757561613430185

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