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Properties

Label 2-280-5.4-c1-0-2
Degree $2$
Conductor $280$
Sign $-0.590 - 0.807i$
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  + 3.12i·3-s + (1.32 + 1.80i)5-s + i·7-s − 6.76·9-s + 2.48·11-s − 4.15i·13-s + (−5.64 + 4.12i)15-s − 5.76i·17-s + 1.60·19-s − 3.12·21-s + 7.28i·23-s + (−1.51 + 4.76i)25-s − 11.7i·27-s − 1.45·29-s − 2.24·31-s + ⋯
L(s)  = 1  + 1.80i·3-s + (0.590 + 0.807i)5-s + 0.377i·7-s − 2.25·9-s + 0.749·11-s − 1.15i·13-s + (−1.45 + 1.06i)15-s − 1.39i·17-s + 0.369·19-s − 0.681·21-s + 1.51i·23-s + (−0.303 + 0.952i)25-s − 2.26i·27-s − 0.270·29-s − 0.404·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.590 - 0.807i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.590 - 0.807i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (169, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.590 - 0.807i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.612289 + 1.20637i\)
\(L(\frac12)\) \(\approx\) \(0.612289 + 1.20637i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.32 - 1.80i)T \)
7 \( 1 - iT \)
good3 \( 1 - 3.12iT - 3T^{2} \)
11 \( 1 - 2.48T + 11T^{2} \)
13 \( 1 + 4.15iT - 13T^{2} \)
17 \( 1 + 5.76iT - 17T^{2} \)
19 \( 1 - 1.60T + 19T^{2} \)
23 \( 1 - 7.28iT - 23T^{2} \)
29 \( 1 + 1.45T + 29T^{2} \)
31 \( 1 + 2.24T + 31T^{2} \)
37 \( 1 - 6iT - 37T^{2} \)
41 \( 1 - 11.2T + 41T^{2} \)
43 \( 1 + 5.28iT - 43T^{2} \)
47 \( 1 + 3.45iT - 47T^{2} \)
53 \( 1 + 9.21iT - 53T^{2} \)
59 \( 1 - 5.92T + 59T^{2} \)
61 \( 1 - 5.35T + 61T^{2} \)
67 \( 1 - 7.52iT - 67T^{2} \)
71 \( 1 + 4.24T + 71T^{2} \)
73 \( 1 - 7.28iT - 73T^{2} \)
79 \( 1 + 16.9T + 79T^{2} \)
83 \( 1 + 10.1iT - 83T^{2} \)
89 \( 1 - 11.4T + 89T^{2} \)
97 \( 1 - 2.73iT - 97T^{2} \)
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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

−11.66430164354522616515318200997, −11.18285032089813547904340895736, −10.05060220758935175064394114861, −9.659499718052433236410763549791, −8.785671283976518353775077611550, −7.30520337107704749538581028257, −5.79582057990735480198262659015, −5.18147894339471837103817592521, −3.70514806944754872798961429427, −2.82594902690966833948318063002, 1.15659279637998945377542962898, 2.18492377511569047977755974177, 4.25260520515035561343464319951, 5.92086896655775988998790677597, 6.51388293268700745537239985374, 7.56176233477994963597965409239, 8.567719002160577968354024830508, 9.296354856936306537048281077586, 10.82213736567031274633330293819, 11.87225815769509691675535403606

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