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Properties

Label 2-720-144.11-c1-0-19
Degree $2$
Conductor $720$
Sign $0.556 - 0.831i$
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  + (−1.35 − 0.414i)2-s + (0.970 − 1.43i)3-s + (1.65 + 1.12i)4-s + (0.965 + 0.258i)5-s + (−1.90 + 1.53i)6-s + (−2.03 + 3.51i)7-s + (−1.77 − 2.20i)8-s + (−1.11 − 2.78i)9-s + (−1.19 − 0.750i)10-s + (−5.55 + 1.48i)11-s + (3.21 − 1.28i)12-s + (2.41 + 0.646i)13-s + (4.20 − 3.91i)14-s + (1.30 − 1.13i)15-s + (1.48 + 3.71i)16-s + 6.82i·17-s + ⋯
L(s)  = 1  + (−0.956 − 0.293i)2-s + (0.560 − 0.828i)3-s + (0.828 + 0.560i)4-s + (0.431 + 0.115i)5-s + (−0.778 + 0.627i)6-s + (−0.767 + 1.32i)7-s + (−0.627 − 0.778i)8-s + (−0.372 − 0.928i)9-s + (−0.379 − 0.237i)10-s + (−1.67 + 0.448i)11-s + (0.928 − 0.372i)12-s + (0.669 + 0.179i)13-s + (1.12 − 1.04i)14-s + (0.337 − 0.293i)15-s + (0.372 + 0.928i)16-s + 1.65i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.556 - 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.556 - 0.831i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (11, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.556 - 0.831i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.748633 + 0.399792i\)
\(L(\frac12)\) \(\approx\) \(0.748633 + 0.399792i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.35 + 0.414i)T \)
3 \( 1 + (-0.970 + 1.43i)T \)
5 \( 1 + (-0.965 - 0.258i)T \)
good7 \( 1 + (2.03 - 3.51i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (5.55 - 1.48i)T + (9.52 - 5.5i)T^{2} \)
13 \( 1 + (-2.41 - 0.646i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 - 6.82iT - 17T^{2} \)
19 \( 1 + (-2.74 - 2.74i)T + 19iT^{2} \)
23 \( 1 + (-5.50 + 3.17i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.30 - 0.618i)T + (25.1 - 14.5i)T^{2} \)
31 \( 1 + (7.96 - 4.60i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + (-4.82 - 4.82i)T + 37iT^{2} \)
41 \( 1 + (1.50 + 2.60i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.88 - 7.04i)T + (-37.2 + 21.5i)T^{2} \)
47 \( 1 + (0.945 - 1.63i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.91 + 5.91i)T - 53iT^{2} \)
59 \( 1 + (1.48 - 5.52i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.763 + 2.85i)T + (-52.8 + 30.5i)T^{2} \)
67 \( 1 + (0.926 - 3.45i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + 9.06iT - 71T^{2} \)
73 \( 1 - 5.63iT - 73T^{2} \)
79 \( 1 + (1.90 + 1.09i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.67 - 6.24i)T + (-71.8 + 41.5i)T^{2} \)
89 \( 1 + 8.98T + 89T^{2} \)
97 \( 1 + (-3.72 + 6.45i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.38058738476862515231839007289, −9.545459852788137736177613275609, −8.748227572970768538249180014329, −8.205629555953991963959551808717, −7.24611685749105707239396575848, −6.27820260327151403534406231973, −5.60602286315211062720628684416, −3.36878507443624667311318858858, −2.60032438999561159652953077016, −1.65687244469494422749771161002, 0.53664949548394235303685265258, 2.58235750801494069086418836721, 3.43074681201309831104305234395, 5.02148990047019786158210829367, 5.74903043599129314195651902302, 7.34523280822677820892052892630, 7.48393253905342482508620868077, 8.807770715004557227846007524118, 9.465886487015866146307361244991, 10.06750022090590537110265389234

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