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Properties

Label 2-280-35.3-c1-0-1
Degree $2$
Conductor $280$
Sign $0.206 - 0.978i$
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  + (0.0322 + 0.120i)3-s + (−1.80 + 1.31i)5-s + (0.969 + 2.46i)7-s + (2.58 − 1.49i)9-s + (−1.40 + 2.44i)11-s + (−3.28 + 3.28i)13-s + (−0.216 − 0.174i)15-s + (1.78 − 0.477i)17-s + (4.01 + 6.95i)19-s + (−0.264 + 0.195i)21-s + (−0.617 + 2.30i)23-s + (1.53 − 4.75i)25-s + (0.526 + 0.526i)27-s − 8.63i·29-s + (−2.81 − 1.62i)31-s + ⋯
L(s)  = 1  + (0.0186 + 0.0694i)3-s + (−0.808 + 0.588i)5-s + (0.366 + 0.930i)7-s + (0.861 − 0.497i)9-s + (−0.424 + 0.736i)11-s + (−0.911 + 0.911i)13-s + (−0.0559 − 0.0451i)15-s + (0.431 − 0.115i)17-s + (0.921 + 1.59i)19-s + (−0.0577 + 0.0427i)21-s + (−0.128 + 0.480i)23-s + (0.306 − 0.951i)25-s + (0.101 + 0.101i)27-s − 1.60i·29-s + (−0.505 − 0.291i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.206 - 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.206 - 0.978i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.206 - 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.862431 + 0.699730i\)
\(L(\frac12)\) \(\approx\) \(0.862431 + 0.699730i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.80 - 1.31i)T \)
7 \( 1 + (-0.969 - 2.46i)T \)
good3 \( 1 + (-0.0322 - 0.120i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.40 - 2.44i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.28 - 3.28i)T - 13iT^{2} \)
17 \( 1 + (-1.78 + 0.477i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-4.01 - 6.95i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (0.617 - 2.30i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 8.63iT - 29T^{2} \)
31 \( 1 + (2.81 + 1.62i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.99 - 1.87i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + 9.45iT - 41T^{2} \)
43 \( 1 + (-1.04 - 1.04i)T + 43iT^{2} \)
47 \( 1 + (1.00 - 3.76i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.54 - 1.75i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-6.19 + 10.7i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-2.19 + 1.26i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.71 + 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 - 2.72T + 71T^{2} \)
73 \( 1 + (-1.04 - 3.90i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-5.52 + 3.18i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (7.41 - 7.41i)T - 83iT^{2} \)
89 \( 1 + (-0.487 - 0.844i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-5.12 - 5.12i)T + 97iT^{2} \)
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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

−12.08859548781782398236974247502, −11.35259682669528620335367986654, −9.962671364084066460017040308878, −9.489837601868495811864623012588, −7.912370697046731857806349093720, −7.41854111316878885779558480564, −6.14710365432777654118844789942, −4.79368241585160898520620904971, −3.69230985412074088929996405847, −2.12736422951316063438072961172, 0.904687185649410155191357311141, 3.12487168889228225093968636094, 4.53391926605488339151219047885, 5.24070821800709221262823628331, 7.16528147843702737127390422582, 7.62489669930325785676871394419, 8.606194230596627898469163679790, 9.895526536280873834205234163921, 10.78126022293941584329812134882, 11.55412788941637231755246233344

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