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Properties

Label 2-280-35.33-c1-0-6
Degree $2$
Conductor $280$
Sign $0.852 - 0.523i$
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  + (2.19 + 0.588i)3-s + (−1.19 + 1.88i)5-s + (1.40 − 2.24i)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 + (−1.34 + 5.00i)17-s + (−0.687 + 1.19i)19-s + (4.40 − 4.10i)21-s + (−4.00 + 1.07i)23-s + (−2.12 − 4.52i)25-s + (−1.32 − 1.32i)27-s − 9.39i·29-s + (−6.08 + 3.51i)31-s + ⋯
L(s)  = 1  + (1.26 + 0.340i)3-s + (−0.536 + 0.844i)5-s + (0.530 − 0.847i)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 + (−0.325 + 1.21i)17-s + (−0.157 + 0.273i)19-s + (0.961 − 0.895i)21-s + (−0.835 + 0.223i)23-s + (−0.425 − 0.905i)25-s + (−0.254 − 0.254i)27-s − 1.74i·29-s + (−1.09 + 0.630i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.852 - 0.523i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.852 - 0.523i$
Analytic conductor: \(2.23581\)
Root analytic conductor: \(1.49526\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{280} (33, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 280,\ (\ :1/2),\ 0.852 - 0.523i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.74642 + 0.493330i\)
\(L(\frac12)\) \(\approx\) \(1.74642 + 0.493330i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 1.88i)T \)
7 \( 1 + (-1.40 + 2.24i)T \)
good3 \( 1 + (-2.19 - 0.588i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (-1.78 - 3.09i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-3.13 + 3.13i)T - 13iT^{2} \)
17 \( 1 + (1.34 - 5.00i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.687 - 1.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (4.00 - 1.07i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 + 9.39iT - 29T^{2} \)
31 \( 1 + (6.08 - 3.51i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.68 + 6.29i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 4.63iT - 41T^{2} \)
43 \( 1 + (-2.51 - 2.51i)T + 43iT^{2} \)
47 \( 1 + (-8.76 + 2.34i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.470 - 1.75i)T + (-45.8 - 26.5i)T^{2} \)
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 \( 1 + (4.37 + 1.17i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 10.5T + 71T^{2} \)
73 \( 1 + (-2.44 - 0.654i)T + (63.2 + 36.5i)T^{2} \)
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

−11.87054888214857169050899214441, −10.70339270973247993767968796610, −10.21134568897799897984496122826, −8.968876544388731073867756977727, −7.986639738555875973116082489107, −7.44840334892032758916534475031, −6.10392840583362738545893757811, −4.05809328334975758667296726279, −3.73152143627886939848096279671, −2.10163300573331531316442339363, 1.64529608170274148571938404233, 3.12643957805687919002386385980, 4.34832812303559429032388258796, 5.68981845416977532181853682452, 7.14125116601134650815903997046, 8.254395358303090858229153348326, 8.865432754656495232754240751820, 9.221430157466288341557696055161, 11.16345112802570445899262945248, 11.75913037313035329417652903986

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