Properties

Label 2-280-35.12-c1-0-2
Degree $2$
Conductor $280$
Sign $0.322 - 0.946i$
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  + (−0.364 + 1.35i)3-s + (−0.489 − 2.18i)5-s + (−0.501 + 2.59i)7-s + (0.882 + 0.509i)9-s + (1.86 + 3.23i)11-s + (4.55 + 4.55i)13-s + (3.14 + 0.129i)15-s + (−5.91 − 1.58i)17-s + (0.616 − 1.06i)19-s + (−3.34 − 1.62i)21-s + (0.721 + 2.69i)23-s + (−4.52 + 2.13i)25-s + (−3.99 + 3.99i)27-s − 2.82i·29-s + (5.94 − 3.43i)31-s + ⋯
L(s)  = 1  + (−0.210 + 0.784i)3-s + (−0.218 − 0.975i)5-s + (−0.189 + 0.981i)7-s + (0.294 + 0.169i)9-s + (0.563 + 0.975i)11-s + (1.26 + 1.26i)13-s + (0.811 + 0.0333i)15-s + (−1.43 − 0.384i)17-s + (0.141 − 0.244i)19-s + (−0.730 − 0.355i)21-s + (0.150 + 0.561i)23-s + (−0.904 + 0.427i)25-s + (−0.769 + 0.769i)27-s − 0.525i·29-s + (1.06 − 0.616i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.322 - 0.946i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(280\)    =    \(2^{3} \cdot 5 \cdot 7\)
Sign: $0.322 - 0.946i$
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: \((2,\ 280,\ (\ :1/2),\ 0.322 - 0.946i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.951415 + 0.680899i\)
\(L(\frac12)\) \(\approx\) \(0.951415 + 0.680899i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.489 + 2.18i)T \)
7 \( 1 + (0.501 - 2.59i)T \)
good3 \( 1 + (0.364 - 1.35i)T + (-2.59 - 1.5i)T^{2} \)
11 \( 1 + (-1.86 - 3.23i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.55 - 4.55i)T + 13iT^{2} \)
17 \( 1 + (5.91 + 1.58i)T + (14.7 + 8.5i)T^{2} \)
19 \( 1 + (-0.616 + 1.06i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-0.721 - 2.69i)T + (-19.9 + 11.5i)T^{2} \)
29 \( 1 + 2.82iT - 29T^{2} \)
31 \( 1 + (-5.94 + 3.43i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-7.52 + 2.01i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + 5.56iT - 41T^{2} \)
43 \( 1 + (1.95 - 1.95i)T - 43iT^{2} \)
47 \( 1 + (2.98 + 11.1i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-9.44 - 2.53i)T + (45.8 + 26.5i)T^{2} \)
59 \( 1 + (-0.916 - 1.58i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (3.43 + 1.98i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (2.12 - 7.93i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 0.570T + 71T^{2} \)
73 \( 1 + (0.546 - 2.03i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (9.47 + 5.47i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (8.80 + 8.80i)T + 83iT^{2} \)
89 \( 1 + (-3.00 + 5.20i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (3.70 - 3.70i)T - 97iT^{2} \)
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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

−11.72245744206586662965861273385, −11.43851170074421311468739826660, −9.917072548031691146088620753585, −9.172424609608034681614342378597, −8.631004751683517967543759503060, −7.07855853997514519958815631539, −5.89830314879483199636745406885, −4.63801585247762892884673765995, −4.08937482648178341937602802444, −1.92724897853119836866160696668, 1.02325293313171568007617336136, 3.10163954149736761816763619683, 4.14357616339056791717272514644, 6.24741308956382659686060551633, 6.51637749235625177827238221415, 7.67210528027738303967224513860, 8.546796026047725482829358033321, 10.07388723629347228429507875542, 10.88674920780437387053288132949, 11.45114443967298720447707225544

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