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Properties

Label 2-280-35.3-c1-0-3
Degree $2$
Conductor $280$
Sign $0.170 - 0.985i$
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  + (0.588 + 2.19i)3-s + (1.03 + 1.98i)5-s + (2.24 − 1.40i)7-s + (−1.88 + 1.08i)9-s + (1.78 − 3.09i)11-s + (−3.13 + 3.13i)13-s + (−3.74 + 3.44i)15-s + (−5.00 + 1.34i)17-s + (0.687 + 1.19i)19-s + (4.40 + 4.10i)21-s + (1.07 − 4.00i)23-s + (−2.85 + 4.10i)25-s + (1.32 + 1.32i)27-s − 9.39i·29-s + (−6.08 − 3.51i)31-s + ⋯
L(s)  = 1  + (0.340 + 1.26i)3-s + (0.462 + 0.886i)5-s + (0.847 − 0.530i)7-s + (−0.628 + 0.363i)9-s + (0.539 − 0.934i)11-s + (−0.869 + 0.869i)13-s + (−0.967 + 0.888i)15-s + (−1.21 + 0.325i)17-s + (0.157 + 0.273i)19-s + (0.961 + 0.895i)21-s + (0.223 − 0.835i)23-s + (−0.571 + 0.820i)25-s + (0.254 + 0.254i)27-s − 1.74i·29-s + (−1.09 − 0.630i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.170 - 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.170 - 0.985i$
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: \((2,\ 280,\ (\ :1/2),\ 0.170 - 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.21582 + 1.02340i\)
\(L(\frac12)\) \(\approx\) \(1.21582 + 1.02340i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.03 - 1.98i)T \)
7 \( 1 + (-2.24 + 1.40i)T \)
good3 \( 1 + (-0.588 - 2.19i)T + (-2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.78 + 3.09i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (3.13 - 3.13i)T - 13iT^{2} \)
17 \( 1 + (5.00 - 1.34i)T + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-0.687 - 1.19i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (-1.07 + 4.00i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.39iT - 29T^{2} \)
31 \( 1 + (6.08 + 3.51i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-6.29 - 1.68i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 - 4.63iT - 41T^{2} \)
43 \( 1 + (-2.51 - 2.51i)T + 43iT^{2} \)
47 \( 1 + (-2.34 + 8.76i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.75 + 0.470i)T + (45.8 - 26.5i)T^{2} \)
59 \( 1 + (-1.42 + 2.46i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1.91 + 1.10i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.17 - 4.37i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-0.654 - 2.44i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (-4.82 + 2.78i)T + (39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.863 - 0.863i)T - 83iT^{2} \)
89 \( 1 + (0.430 + 0.745i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (12.6 + 12.6i)T + 97iT^{2} \)
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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

−11.53340625252877826360189167911, −11.08478721727471453484717363042, −10.14662517387211994465711216013, −9.418210763782102938280382970170, −8.450142341786921449754291283748, −7.15840984506624565779132117275, −6.05311022549568555553571822664, −4.59406066251749722873538486243, −3.85108306234074506304954412542, −2.33771468753195203096052225025, 1.43918276607446097290447039855, 2.42528254331250410895062916140, 4.64475262390147886694450928076, 5.54560423640587827714359280121, 6.98811612677787954081044236597, 7.66296671113694577178765192375, 8.771720413729485059949279449865, 9.393616929003181283032303920515, 10.85700670851832482740531033283, 12.09026463656796344440109331766

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