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Properties

Label 2-720-15.8-c3-0-14
Degree $2$
Conductor $720$
Sign $0.314 - 0.949i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (8.38 + 7.38i)5-s + (10.8 − 10.8i)7-s + 37.8i·11-s + (−48.1 − 48.1i)13-s + (60.3 + 60.3i)17-s + 109. i·19-s + (39.7 − 39.7i)23-s + (15.7 + 123. i)25-s − 90.3·29-s + 233.·31-s + (171. − 10.8i)35-s + (−19.0 + 19.0i)37-s + 260. i·41-s + (−176. − 176. i)43-s + (−145. − 145. i)47-s + ⋯
L(s)  = 1  + (0.750 + 0.660i)5-s + (0.586 − 0.586i)7-s + 1.03i·11-s + (−1.02 − 1.02i)13-s + (0.860 + 0.860i)17-s + 1.32i·19-s + (0.360 − 0.360i)23-s + (0.126 + 0.991i)25-s − 0.578·29-s + 1.35·31-s + (0.828 − 0.0524i)35-s + (−0.0844 + 0.0844i)37-s + 0.993i·41-s + (−0.624 − 0.624i)43-s + (−0.452 − 0.452i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.314 - 0.949i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.314 - 0.949i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 0.314 - 0.949i)\)

Particular Values

\(L(2)\) \(\approx\) \(2.265264327\)
\(L(\frac12)\) \(\approx\) \(2.265264327\)
\(L(\frac{5}{2})\) 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 \( 1 \)
5 \( 1 + (-8.38 - 7.38i)T \)
good7 \( 1 + (-10.8 + 10.8i)T - 343iT^{2} \)
11 \( 1 - 37.8iT - 1.33e3T^{2} \)
13 \( 1 + (48.1 + 48.1i)T + 2.19e3iT^{2} \)
17 \( 1 + (-60.3 - 60.3i)T + 4.91e3iT^{2} \)
19 \( 1 - 109. iT - 6.85e3T^{2} \)
23 \( 1 + (-39.7 + 39.7i)T - 1.21e4iT^{2} \)
29 \( 1 + 90.3T + 2.43e4T^{2} \)
31 \( 1 - 233.T + 2.97e4T^{2} \)
37 \( 1 + (19.0 - 19.0i)T - 5.06e4iT^{2} \)
41 \( 1 - 260. iT - 6.89e4T^{2} \)
43 \( 1 + (176. + 176. i)T + 7.95e4iT^{2} \)
47 \( 1 + (145. + 145. i)T + 1.03e5iT^{2} \)
53 \( 1 + (-183. + 183. i)T - 1.48e5iT^{2} \)
59 \( 1 - 279.T + 2.05e5T^{2} \)
61 \( 1 + 390.T + 2.26e5T^{2} \)
67 \( 1 + (150. - 150. i)T - 3.00e5iT^{2} \)
71 \( 1 - 470. iT - 3.57e5T^{2} \)
73 \( 1 + (-480. - 480. i)T + 3.89e5iT^{2} \)
79 \( 1 - 1.32e3iT - 4.93e5T^{2} \)
83 \( 1 + (-456. + 456. i)T - 5.71e5iT^{2} \)
89 \( 1 - 1.36e3T + 7.04e5T^{2} \)
97 \( 1 + (785. - 785. i)T - 9.12e5iT^{2} \)
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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

−10.09418219263475551561131504475, −9.779530493805257047710340356692, −8.226117516212122055537330746063, −7.61311706208841146180839374029, −6.73303077166377962656655083726, −5.68588657398687255331598575417, −4.83404929201779430753890567174, −3.60263388413848132097660582666, −2.39379477035033710238943359754, −1.30362294025352802985134748179, 0.65016325259040332029217822414, 1.96609703172301409086136333664, 2.99830000849697388716203356765, 4.67971186961639932349197794380, 5.19238678956248620134443284672, 6.17345636089180963646326113902, 7.22727341646705907272450125560, 8.278604308431431697230306430863, 9.145939582479844186027656587197, 9.538427173394511135349876697893

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