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Properties

Label 2-280-280.139-c1-0-2
Degree $2$
Conductor $280$
Sign $0.863 - 0.504i$
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  + (−0.244 − 1.39i)2-s − 2.91·3-s + (−1.88 + 0.682i)4-s + (−1.83 − 1.27i)5-s + (0.712 + 4.05i)6-s + (−1.52 − 2.16i)7-s + (1.41 + 2.45i)8-s + 5.47·9-s + (−1.32 + 2.87i)10-s − 0.0929·11-s + (5.47 − 1.98i)12-s + 4.08i·13-s + (−2.64 + 2.65i)14-s + (5.34 + 3.71i)15-s + (3.06 − 2.56i)16-s + 4.24·17-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s − 1.68·3-s + (−0.940 + 0.341i)4-s + (−0.821 − 0.570i)5-s + (0.290 + 1.65i)6-s + (−0.575 − 0.817i)7-s + (0.498 + 0.866i)8-s + 1.82·9-s + (−0.419 + 0.907i)10-s − 0.0280·11-s + (1.57 − 0.573i)12-s + 1.13i·13-s + (−0.705 + 0.708i)14-s + (1.38 + 0.958i)15-s + (0.767 − 0.641i)16-s + 1.02·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.863 - 0.504i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.863 - 0.504i$
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: \((2,\ 280,\ (\ :1/2),\ 0.863 - 0.504i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.211527 + 0.0572053i\)
\(L(\frac12)\) \(\approx\) \(0.211527 + 0.0572053i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.244 + 1.39i)T \)
5 \( 1 + (1.83 + 1.27i)T \)
7 \( 1 + (1.52 + 2.16i)T \)
good3 \( 1 + 2.91T + 3T^{2} \)
11 \( 1 + 0.0929T + 11T^{2} \)
13 \( 1 - 4.08iT - 13T^{2} \)
17 \( 1 - 4.24T + 17T^{2} \)
19 \( 1 - 2.39iT - 19T^{2} \)
23 \( 1 + 4.52T + 23T^{2} \)
29 \( 1 + 4.35iT - 29T^{2} \)
31 \( 1 + 1.10T + 31T^{2} \)
37 \( 1 + 8.54T + 37T^{2} \)
41 \( 1 - 6.10iT - 41T^{2} \)
43 \( 1 - 4.60iT - 43T^{2} \)
47 \( 1 - 7.93iT - 47T^{2} \)
53 \( 1 - 14.3T + 53T^{2} \)
59 \( 1 + 10.6iT - 59T^{2} \)
61 \( 1 + 1.92T + 61T^{2} \)
67 \( 1 - 13.1iT - 67T^{2} \)
71 \( 1 - 9.75iT - 71T^{2} \)
73 \( 1 + 6.19T + 73T^{2} \)
79 \( 1 - 3.42iT - 79T^{2} \)
83 \( 1 - 11.7T + 83T^{2} \)
89 \( 1 - 4.46iT - 89T^{2} \)
97 \( 1 + 10.1T + 97T^{2} \)
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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

−11.88608446982726706755097189878, −11.20055501490118015459139883744, −10.26274825223881537841967950208, −9.571021527014235649430742535921, −8.096670972755696304387770298707, −7.01899763866964080393837797777, −5.70244812736885082329691634336, −4.51985119397586927955414445096, −3.79355739518682831272467570798, −1.17820231855151035510197697521, 0.26231585147456051685581806016, 3.62812284106588593630653517678, 5.17261866260460790795207138318, 5.75843164359202269428433262091, 6.74686036767882908308082819338, 7.54303987325615735146443215618, 8.756644250327616168774719198697, 10.20102582402981477476787608754, 10.61310146650574375480037492359, 12.09566566492904618323810728518

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