Properties

Label 2-280-35.12-c1-0-9
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: $\chi_{280} (257, \cdot )$
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

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

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