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Properties

Label 2-720-45.2-c1-0-4
Degree $2$
Conductor $720$
Sign $-0.976 - 0.214i$
Analytic cond. $5.74922$
Root an. cond. $2.39775$
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  + (1.66 + 0.483i)3-s + (−1.17 + 1.90i)5-s + (−4.62 + 1.23i)7-s + (2.53 + 1.60i)9-s + (−4.22 − 2.43i)11-s + (−1.04 − 0.281i)13-s + (−2.87 + 2.59i)15-s + (−0.519 + 0.519i)17-s − 1.23i·19-s + (−8.29 − 0.175i)21-s + (−0.736 + 2.74i)23-s + (−2.24 − 4.46i)25-s + (3.43 + 3.89i)27-s + (−1.85 + 3.20i)29-s + (−2.78 − 4.83i)31-s + ⋯
L(s)  = 1  + (0.960 + 0.279i)3-s + (−0.525 + 0.851i)5-s + (−1.74 + 0.468i)7-s + (0.844 + 0.536i)9-s + (−1.27 − 0.735i)11-s + (−0.290 − 0.0779i)13-s + (−0.741 + 0.670i)15-s + (−0.126 + 0.126i)17-s − 0.284i·19-s + (−1.80 − 0.0382i)21-s + (−0.153 + 0.572i)23-s + (−0.448 − 0.893i)25-s + (0.660 + 0.750i)27-s + (−0.343 + 0.595i)29-s + (−0.501 − 0.867i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.976 - 0.214i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.976 - 0.214i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (497, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.976 - 0.214i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0715718 + 0.660202i\)
\(L(\frac12)\) \(\approx\) \(0.0715718 + 0.660202i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.66 - 0.483i)T \)
5 \( 1 + (1.17 - 1.90i)T \)
good7 \( 1 + (4.62 - 1.23i)T + (6.06 - 3.5i)T^{2} \)
11 \( 1 + (4.22 + 2.43i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.04 + 0.281i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (0.519 - 0.519i)T - 17iT^{2} \)
19 \( 1 + 1.23iT - 19T^{2} \)
23 \( 1 + (0.736 - 2.74i)T + (-19.9 - 11.5i)T^{2} \)
29 \( 1 + (1.85 - 3.20i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.78 + 4.83i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-1.66 - 1.66i)T + 37iT^{2} \)
41 \( 1 + (5.59 - 3.23i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-2.77 - 10.3i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (-1.43 - 5.36i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.12 - 2.12i)T + 53iT^{2} \)
59 \( 1 + (-4.83 - 8.37i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-1.49 + 2.59i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-3.03 + 11.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 - 14.8iT - 71T^{2} \)
73 \( 1 + (5.57 - 5.57i)T - 73iT^{2} \)
79 \( 1 + (4.35 + 2.51i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.48 + 0.933i)T + (71.8 - 41.5i)T^{2} \)
89 \( 1 + 1.54T + 89T^{2} \)
97 \( 1 + (12.5 - 3.35i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.58072190768359741819703077127, −9.896579676607707557830538247908, −9.214208098928096585547372928396, −8.186211861458478147914503479398, −7.44328852351550427900436850260, −6.54026514053937036088768913399, −5.52748443663074473000762966909, −4.00946218397550491640566775287, −3.04008552186293094630502934195, −2.65539406774115824157634034534, 0.28084264331013515957745366787, 2.23292815473674135328136396870, 3.38910879257836567452994626900, 4.20925968908779274162810565602, 5.43903398921314344486695140555, 6.84584888204251148147912360253, 7.37593415514863147864499025400, 8.315772848334311388387580065523, 9.148934947086049647528347596018, 9.902986453406669460565424992252

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