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Properties

Label 2-280-35.9-c1-0-7
Degree $2$
Conductor $280$
Sign $0.741 + 0.671i$
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  + (1.82 − 1.05i)3-s + (0.728 − 2.11i)5-s + (1.44 + 2.21i)7-s + (0.722 − 1.25i)9-s + (−0.887 − 1.53i)11-s + 1.44i·13-s + (−0.898 − 4.62i)15-s + (−5.08 + 2.93i)17-s + (3.58 − 6.20i)19-s + (4.97 + 2.51i)21-s + (−0.574 − 0.331i)23-s + (−3.93 − 3.08i)25-s + 3.27i·27-s − 6.45·29-s + (5.03 + 8.72i)31-s + ⋯
L(s)  = 1  + (1.05 − 0.608i)3-s + (0.325 − 0.945i)5-s + (0.547 + 0.836i)7-s + (0.240 − 0.417i)9-s + (−0.267 − 0.463i)11-s + 0.400i·13-s + (−0.231 − 1.19i)15-s + (−1.23 + 0.711i)17-s + (0.821 − 1.42i)19-s + (1.08 + 0.548i)21-s + (−0.119 − 0.0691i)23-s + (−0.787 − 0.616i)25-s + 0.631i·27-s − 1.19·29-s + (0.904 + 1.56i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.741 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.741 + 0.671i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.741 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.71546 - 0.661395i\)
\(L(\frac12)\) \(\approx\) \(1.71546 - 0.661395i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.728 + 2.11i)T \)
7 \( 1 + (-1.44 - 2.21i)T \)
good3 \( 1 + (-1.82 + 1.05i)T + (1.5 - 2.59i)T^{2} \)
11 \( 1 + (0.887 + 1.53i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 - 1.44iT - 13T^{2} \)
17 \( 1 + (5.08 - 2.93i)T + (8.5 - 14.7i)T^{2} \)
19 \( 1 + (-3.58 + 6.20i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (0.574 + 0.331i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.45T + 29T^{2} \)
31 \( 1 + (-5.03 - 8.72i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-4.34 - 2.50i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + 1.92T + 41T^{2} \)
43 \( 1 + 5.81iT - 43T^{2} \)
47 \( 1 + (-3.20 - 1.85i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.513 + 0.296i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.79 + 6.57i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.36 - 7.55i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1.87 - 1.08i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6.49T + 71T^{2} \)
73 \( 1 + (13.4 - 7.76i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (1.35 - 2.34i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 6.35iT - 83T^{2} \)
89 \( 1 + (-2.93 + 5.07i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + 14.7iT - 97T^{2} \)
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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

−11.89466543120118448399915650820, −10.94554268802935884559674854624, −9.373882015439599416462236239986, −8.739590980144627209689795048882, −8.237277851644789748353053692565, −7.00103186248130454246325679374, −5.63438271958630441775240162711, −4.57406067131020830271499649864, −2.79113989873482892837580905791, −1.69247795297779233922124817162, 2.24115109497425570464000653122, 3.45625455583222784581306765506, 4.46356874306677308759146946051, 6.01750423323878721580062951031, 7.40306762378179960531162352345, 7.966829198820784797481383780559, 9.364124025929578719824848324765, 9.962940645378970882913627853364, 10.83090817662582109574279794561, 11.74397650639815487507154151638

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