Properties

Label 2-280-35.33-c1-0-0
Degree $2$
Conductor $280$
Sign $-0.990 - 0.135i$
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  + (−1.16 − 0.312i)3-s + (−0.121 + 2.23i)5-s + (−2.59 − 0.491i)7-s + (−1.33 − 0.770i)9-s + (−1.67 − 2.90i)11-s + (−2.92 + 2.92i)13-s + (0.840 − 2.56i)15-s + (0.0694 − 0.259i)17-s + (−0.458 + 0.793i)19-s + (2.88 + 1.38i)21-s + (−7.58 + 2.03i)23-s + (−4.97 − 0.544i)25-s + (3.87 + 3.87i)27-s + 1.31i·29-s + (3.25 − 1.88i)31-s + ⋯
L(s)  = 1  + (−0.673 − 0.180i)3-s + (−0.0544 + 0.998i)5-s + (−0.982 − 0.185i)7-s + (−0.444 − 0.256i)9-s + (−0.506 − 0.877i)11-s + (−0.810 + 0.810i)13-s + (0.216 − 0.662i)15-s + (0.0168 − 0.0628i)17-s + (−0.105 + 0.182i)19-s + (0.628 + 0.302i)21-s + (−1.58 + 0.423i)23-s + (−0.994 − 0.108i)25-s + (0.746 + 0.746i)27-s + 0.244i·29-s + (0.584 − 0.337i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.990 - 0.135i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $-0.990 - 0.135i$
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: \((2,\ 280,\ (\ :1/2),\ -0.990 - 0.135i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00800376 + 0.117399i\)
\(L(\frac12)\) \(\approx\) \(0.00800376 + 0.117399i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.121 - 2.23i)T \)
7 \( 1 + (2.59 + 0.491i)T \)
good3 \( 1 + (1.16 + 0.312i)T + (2.59 + 1.5i)T^{2} \)
11 \( 1 + (1.67 + 2.90i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.92 - 2.92i)T - 13iT^{2} \)
17 \( 1 + (-0.0694 + 0.259i)T + (-14.7 - 8.5i)T^{2} \)
19 \( 1 + (0.458 - 0.793i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (7.58 - 2.03i)T + (19.9 - 11.5i)T^{2} \)
29 \( 1 - 1.31iT - 29T^{2} \)
31 \( 1 + (-3.25 + 1.88i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.11 - 7.90i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + 5.65iT - 41T^{2} \)
43 \( 1 + (2.61 + 2.61i)T + 43iT^{2} \)
47 \( 1 + (0.911 - 0.244i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-3.49 + 13.0i)T + (-45.8 - 26.5i)T^{2} \)
59 \( 1 + (-3.91 - 6.78i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-9.70 - 5.60i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.37 - 1.44i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 + (7.33 + 1.96i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.87 + 1.08i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (5.49 - 5.49i)T - 83iT^{2} \)
89 \( 1 + (1.34 - 2.33i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (10.1 + 10.1i)T + 97iT^{2} \)
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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

−11.95397900606036234983498916198, −11.56739835138239315228329870403, −10.39516690350320179099335498478, −9.786557514963137011497931115416, −8.427181783041623248803428528912, −7.14916940097486162230107952942, −6.38542809131578359171148196613, −5.58533548503462767514115918440, −3.81614485961214514550032329993, −2.65354313945599222190451924259, 0.088655455158749105155736033102, 2.54019345114765807860554968518, 4.33361059966823359466793177094, 5.32283508686170750002523775752, 6.14734612671133020999772680629, 7.57240351090141854363694996280, 8.527000801951196131451000079722, 9.774341513755168758214986679861, 10.24282388807917355312740812002, 11.58590384254675212806379538573

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