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Properties

Label 2-720-9.7-c1-0-7
Degree $2$
Conductor $720$
Sign $0.669 - 0.742i$
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  + (−1.24 + 1.20i)3-s + (−0.5 − 0.866i)5-s + (0.433 − 0.751i)7-s + (0.106 − 2.99i)9-s + (−2.04 + 3.53i)11-s + (−0.606 − 1.05i)13-s + (1.66 + 0.477i)15-s + 7.82·17-s + 4.51·19-s + (0.363 + 1.45i)21-s + (1.43 + 2.48i)23-s + (−0.499 + 0.866i)25-s + (3.47 + 3.86i)27-s + (−3.14 + 5.45i)29-s + (−1.26 − 2.18i)31-s + ⋯
L(s)  = 1  + (−0.719 + 0.694i)3-s + (−0.223 − 0.387i)5-s + (0.163 − 0.284i)7-s + (0.0354 − 0.999i)9-s + (−0.616 + 1.06i)11-s + (−0.168 − 0.291i)13-s + (0.429 + 0.123i)15-s + 1.89·17-s + 1.03·19-s + (0.0792 + 0.318i)21-s + (0.298 + 0.517i)23-s + (−0.0999 + 0.173i)25-s + (0.668 + 0.743i)27-s + (−0.584 + 1.01i)29-s + (−0.226 − 0.392i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.669 - 0.742i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (241, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.669 - 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02520 + 0.456140i\)
\(L(\frac12)\) \(\approx\) \(1.02520 + 0.456140i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.24 - 1.20i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-0.433 + 0.751i)T + (-3.5 - 6.06i)T^{2} \)
11 \( 1 + (2.04 - 3.53i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (0.606 + 1.05i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 - 7.82T + 17T^{2} \)
19 \( 1 - 4.51T + 19T^{2} \)
23 \( 1 + (-1.43 - 2.48i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.14 - 5.45i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.26 + 2.18i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + 0.523T + 37T^{2} \)
41 \( 1 + (-4.06 - 7.03i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1.91 + 3.31i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (0.695 - 1.20i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 13.9T + 53T^{2} \)
59 \( 1 + (-3.43 - 5.95i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (4.54 - 7.86i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.68 - 2.92i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + 3.21T + 71T^{2} \)
73 \( 1 - 8.60T + 73T^{2} \)
79 \( 1 + (-8.12 + 14.0i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-3.22 + 5.57i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + 10.2T + 89T^{2} \)
97 \( 1 + (-4.38 + 7.59i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.37137018670068958701062583140, −9.857878800632263839601510693321, −9.060058899163934185726812541707, −7.69203850243896529926763202187, −7.24781106575464843057126152876, −5.66435080284047584293952095219, −5.22455367594052490441799079876, −4.19880811017247485180451306301, −3.13403488459875366128871757761, −1.12772665635971404921131335537, 0.818630129361627111121185963458, 2.45668863991623077475538758715, 3.63758793894270368950913091115, 5.29226933147222312137283044836, 5.67114673179068191656823509709, 6.81254760956637377624472798150, 7.69597236915336566866612265879, 8.248634912695715232770473399527, 9.561781027497790010564085801572, 10.51349887622205689977566682087

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