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Properties

Label 2-720-9.4-c1-0-0
Degree $2$
Conductor $720$
Sign $-0.993 + 0.112i$
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  + (−0.5 + 1.65i)3-s + (0.5 − 0.866i)5-s + (−1.18 − 2.05i)7-s + (−2.5 − 1.65i)9-s + (0.686 + 1.18i)11-s + (−2.37 + 4.10i)13-s + (1.18 + 1.26i)15-s − 7.37·17-s − 3.37·19-s + (4 − 0.939i)21-s + (−2.18 + 3.78i)23-s + (−0.499 − 0.866i)25-s + (4 − 3.31i)27-s + (2.18 + 3.78i)29-s + (−3.37 + 5.84i)31-s + ⋯
L(s)  = 1  + (−0.288 + 0.957i)3-s + (0.223 − 0.387i)5-s + (−0.448 − 0.776i)7-s + (−0.833 − 0.552i)9-s + (0.206 + 0.358i)11-s + (−0.657 + 1.13i)13-s + (0.306 + 0.325i)15-s − 1.78·17-s − 0.773·19-s + (0.872 − 0.205i)21-s + (−0.455 + 0.789i)23-s + (−0.0999 − 0.173i)25-s + (0.769 − 0.638i)27-s + (0.405 + 0.703i)29-s + (−0.605 + 1.04i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.993 + 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.993 + 0.112i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.993 + 0.112i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0145419 - 0.258374i\)
\(L(\frac12)\) \(\approx\) \(0.0145419 - 0.258374i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.5 - 1.65i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.18 + 2.05i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-0.686 - 1.18i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (2.37 - 4.10i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 7.37T + 17T^{2} \)
19 \( 1 + 3.37T + 19T^{2} \)
23 \( 1 + (2.18 - 3.78i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.18 - 3.78i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (3.37 - 5.84i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 4T + 37T^{2} \)
41 \( 1 + (-1.5 + 2.59i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (5.68 + 9.84i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (0.813 + 1.40i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 11.4T + 53T^{2} \)
59 \( 1 + (0.686 - 1.18i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (4.55 + 7.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.5 - 6.06i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 14.1T + 73T^{2} \)
79 \( 1 + (-1 - 1.73i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-0.813 - 1.40i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 1.11T + 89T^{2} \)
97 \( 1 + (-1.31 - 2.27i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.62002048394349978183160997564, −10.11341665644614651879082016532, −9.077704936794349875450270548390, −8.759111706272774043867725639257, −7.07883326663211550189425493882, −6.56732739935333797512001845144, −5.25303668978348189925771308335, −4.41517139960761881157285576241, −3.71570529812278272961330773073, −2.06050521472812793312763179561, 0.12636429227923717942868969240, 2.16346493969925411353803671666, 2.86792122337565813396102077544, 4.56292427369574977162951961659, 5.87104830299926570490839906389, 6.28316653760480924353263066762, 7.23150275299571797326843847594, 8.250341870334197249997585534815, 8.926750086812180436451487540851, 10.06297606813231708892529720717

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