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Properties

Label 2-720-9.4-c1-0-1
Degree $2$
Conductor $720$
Sign $0.156 - 0.987i$
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  + (−1.62 + 0.606i)3-s + (−0.5 + 0.866i)5-s + (−2.62 − 4.54i)7-s + (2.26 − 1.96i)9-s + (1.33 + 2.31i)11-s + (−1.90 + 3.30i)13-s + (0.285 − 1.70i)15-s + 3.52·17-s + 4.67·19-s + (7.00 + 5.77i)21-s + (−2.47 + 4.29i)23-s + (−0.499 − 0.866i)25-s + (−2.48 + 4.56i)27-s + (0.928 + 1.60i)29-s + (−4.33 + 7.51i)31-s + ⋯
L(s)  = 1  + (−0.936 + 0.350i)3-s + (−0.223 + 0.387i)5-s + (−0.991 − 1.71i)7-s + (0.754 − 0.655i)9-s + (0.402 + 0.697i)11-s + (−0.529 + 0.916i)13-s + (0.0738 − 0.441i)15-s + 0.855·17-s + 1.07·19-s + (1.52 + 1.26i)21-s + (−0.516 + 0.895i)23-s + (−0.0999 − 0.173i)25-s + (−0.477 + 0.878i)27-s + (0.172 + 0.298i)29-s + (−0.778 + 1.34i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.156 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.156 - 0.987i$
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: \((2,\ 720,\ (\ :1/2),\ 0.156 - 0.987i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.562568 + 0.480407i\)
\(L(\frac12)\) \(\approx\) \(0.562568 + 0.480407i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.62 - 0.606i)T \)
5 \( 1 + (0.5 - 0.866i)T \)
good7 \( 1 + (2.62 + 4.54i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1.33 - 2.31i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (1.90 - 3.30i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 - 3.52T + 17T^{2} \)
19 \( 1 - 4.67T + 19T^{2} \)
23 \( 1 + (2.47 - 4.29i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-0.928 - 1.60i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4.33 - 7.51i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.67T + 37T^{2} \)
41 \( 1 + (-1.83 + 3.18i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-1.76 - 3.05i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-4.63 - 8.02i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 2.85T + 53T^{2} \)
59 \( 1 + (-2.10 + 3.63i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-3.98 - 6.89i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (0.429 - 0.744i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 15.1T + 71T^{2} \)
73 \( 1 - 6.28T + 73T^{2} \)
79 \( 1 + (-2.81 - 4.87i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.94 + 3.37i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 11T + 89T^{2} \)
97 \( 1 + (-1.91 - 3.32i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.54088124367219032180106635577, −9.811924269773780190661843481669, −9.422555225906732314252550309069, −7.45185032219471056680901855173, −7.15896256975885583863930632287, −6.35689961955670522157387141741, −5.11888650745723188652215280235, −4.06354944247541279540910759735, −3.43748869845964402524403569668, −1.19876767267005210879708857616, 0.50219948432079023988840303533, 2.36611698168350663035456654687, 3.57786176758736355667604805600, 5.21256402981575272709082779579, 5.68262608406109503232872281133, 6.41697395840952463008876032048, 7.61115533342594053091684292560, 8.476892435523554835357294410172, 9.486741804083413297862776058404, 10.10358168609639259639121143978

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