Properties

Label 2-280-7.4-c1-0-6
Degree $2$
Conductor $280$
Sign $-0.198 + 0.980i$
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  + (1.20 − 2.09i)3-s + (−0.5 − 0.866i)5-s + (−2.62 − 0.358i)7-s + (−1.41 − 2.44i)9-s + (2.41 − 4.18i)11-s + 2·13-s − 2.41·15-s + (−1.82 + 3.16i)17-s + (−2.82 − 4.89i)19-s + (−3.91 + 5.04i)21-s + (4.20 + 7.28i)23-s + (−0.499 + 0.866i)25-s + 0.414·27-s − 2.17·29-s + (2.41 − 4.18i)31-s + ⋯
L(s)  = 1  + (0.696 − 1.20i)3-s + (−0.223 − 0.387i)5-s + (−0.990 − 0.135i)7-s + (−0.471 − 0.816i)9-s + (0.727 − 1.26i)11-s + 0.554·13-s − 0.623·15-s + (−0.443 + 0.768i)17-s + (−0.648 − 1.12i)19-s + (−0.854 + 1.10i)21-s + (0.877 + 1.51i)23-s + (−0.0999 + 0.173i)25-s + 0.0797·27-s − 0.403·29-s + (0.433 − 0.751i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.198 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.198 + 0.980i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ -0.198 + 0.980i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.886160 - 1.08312i\)
\(L(\frac12)\) \(\approx\) \(0.886160 - 1.08312i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (2.62 + 0.358i)T \)
good3 \( 1 + (-1.20 + 2.09i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (-2.41 + 4.18i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2T + 13T^{2} \)
17 \( 1 + (1.82 - 3.16i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (2.82 + 4.89i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-4.20 - 7.28i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.17T + 29T^{2} \)
31 \( 1 + (-2.41 + 4.18i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.82 - 4.89i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 0.171T + 41T^{2} \)
43 \( 1 - 12.8T + 43T^{2} \)
47 \( 1 + (0.171 + 0.297i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (2.82 - 4.89i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.32 - 4.03i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.44 - 5.97i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 12T + 71T^{2} \)
73 \( 1 + (-3.82 + 6.63i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 13.2T + 83T^{2} \)
89 \( 1 + (-8.32 - 14.4i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 6T + 97T^{2} \)
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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

−11.70776983810600882107548339327, −10.85841089484437965967292738524, −9.245161450829162808671873587279, −8.752062120743038113528435166263, −7.71611618647286109197631231783, −6.69957839915771996295503766977, −5.93320887717007170126663228607, −3.96709219272168696580941660623, −2.82777380518921199031991292441, −1.08428940480459934603475646725, 2.61710245773607030521634850411, 3.79191535636515774799593132944, 4.56465708627764315012572501010, 6.24846153389133326059142244730, 7.20226861002066741457509607574, 8.673342836698844699501629770381, 9.328756323188429715106890515682, 10.12749056726224604454817186653, 10.85747427628714578914166737629, 12.19625132849612388160888549664

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