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Properties

Label 2-280-7.4-c1-0-7
Degree $2$
Conductor $280$
Sign $-0.0391 + 0.999i$
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  + (1.64 − 2.84i)3-s + (−0.5 − 0.866i)5-s + (2.64 − 0.0641i)7-s + (−3.91 − 6.77i)9-s + (−2.91 + 5.04i)11-s + 2.75·13-s − 3.28·15-s + (−1 + 1.73i)17-s + (0.378 + 0.654i)19-s + (4.16 − 7.64i)21-s + (0.266 + 0.462i)23-s + (−0.499 + 0.866i)25-s − 15.8·27-s − 0.823·29-s + (1.28 − 2.23i)31-s + ⋯
L(s)  = 1  + (0.949 − 1.64i)3-s + (−0.223 − 0.387i)5-s + (0.999 − 0.0242i)7-s + (−1.30 − 2.25i)9-s + (−0.877 + 1.52i)11-s + 0.764·13-s − 0.849·15-s + (−0.242 + 0.420i)17-s + (0.0867 + 0.150i)19-s + (0.909 − 1.66i)21-s + (0.0556 + 0.0963i)23-s + (−0.0999 + 0.173i)25-s − 3.05·27-s − 0.152·29-s + (0.231 − 0.401i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0391 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0391 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.0391 + 0.999i$
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: \((2,\ 280,\ (\ :1/2),\ -0.0391 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.17781 - 1.22480i\)
\(L(\frac12)\) \(\approx\) \(1.17781 - 1.22480i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 + 0.0641i)T \)
good3 \( 1 + (-1.64 + 2.84i)T + (-1.5 - 2.59i)T^{2} \)
11 \( 1 + (2.91 - 5.04i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 - 2.75T + 13T^{2} \)
17 \( 1 + (1 - 1.73i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.378 - 0.654i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-0.266 - 0.462i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 0.823T + 29T^{2} \)
31 \( 1 + (-1.28 + 2.23i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (2.37 + 4.11i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 - 6.06T + 41T^{2} \)
43 \( 1 - 0.710T + 43T^{2} \)
47 \( 1 + (-6.44 - 11.1i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.20 + 7.27i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (4 - 6.92i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.70 + 8.14i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.93 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + (1.75 - 3.04i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-4.75 - 8.23i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 6.71T + 83T^{2} \)
89 \( 1 + (0.878 + 1.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + 2T + 97T^{2} \)
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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

−11.98498137111685053028598723890, −10.87066575955017480557838809813, −9.373113542270935384165466833168, −8.406660519840213660207426075253, −7.75589464455358810483602951285, −7.10000538586116551470491137481, −5.73830927318164739903995169283, −4.20934300863081578036029324396, −2.47305310117387144896556057520, −1.44117665930974068488389560900, 2.68431843027983915432498646441, 3.64197973225146673299535902780, 4.76431215049530286584086950921, 5.73394011786525625886312420609, 7.70440293185801771910706542930, 8.491492767484390251738658355465, 9.065258972149391440984182841219, 10.49829720080715462011167959529, 10.79169814450794338390195615560, 11.63579469394524422034157128882

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