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Properties

Label 2-280-35.17-c1-0-9
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: $\chi_{280} (17, \cdot )$
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

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

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