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Properties

Label 2-720-80.69-c1-0-51
Degree $2$
Conductor $720$
Sign $0.186 + 0.982i$
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  + (0.456 + 1.33i)2-s + (−1.58 + 1.22i)4-s + (1.50 − 1.65i)5-s − 2.58·7-s + (−2.35 − 1.55i)8-s + (2.90 + 1.25i)10-s + (−4.39 − 4.39i)11-s + (−0.417 − 0.417i)13-s + (−1.18 − 3.46i)14-s + (1.00 − 3.87i)16-s − 4.40i·17-s + (−4.53 + 4.53i)19-s + (−0.348 + 4.45i)20-s + (3.87 − 7.89i)22-s + 0.281·23-s + ⋯
L(s)  = 1  + (0.323 + 0.946i)2-s + (−0.791 + 0.611i)4-s + (0.671 − 0.741i)5-s − 0.978·7-s + (−0.834 − 0.551i)8-s + (0.918 + 0.395i)10-s + (−1.32 − 1.32i)11-s + (−0.115 − 0.115i)13-s + (−0.316 − 0.926i)14-s + (0.252 − 0.967i)16-s − 1.06i·17-s + (−1.04 + 1.04i)19-s + (−0.0778 + 0.996i)20-s + (0.826 − 1.68i)22-s + 0.0586·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.186 + 0.982i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.186 + 0.982i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (469, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.186 + 0.982i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.524907 - 0.434789i\)
\(L(\frac12)\) \(\approx\) \(0.524907 - 0.434789i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.456 - 1.33i)T \)
3 \( 1 \)
5 \( 1 + (-1.50 + 1.65i)T \)
good7 \( 1 + 2.58T + 7T^{2} \)
11 \( 1 + (4.39 + 4.39i)T + 11iT^{2} \)
13 \( 1 + (0.417 + 0.417i)T + 13iT^{2} \)
17 \( 1 + 4.40iT - 17T^{2} \)
19 \( 1 + (4.53 - 4.53i)T - 19iT^{2} \)
23 \( 1 - 0.281T + 23T^{2} \)
29 \( 1 + (3.73 - 3.73i)T - 29iT^{2} \)
31 \( 1 + 3.05T + 31T^{2} \)
37 \( 1 + (-5.26 + 5.26i)T - 37iT^{2} \)
41 \( 1 - 5.16iT - 41T^{2} \)
43 \( 1 + (-2.66 + 2.66i)T - 43iT^{2} \)
47 \( 1 + 7.45iT - 47T^{2} \)
53 \( 1 + (2.89 - 2.89i)T - 53iT^{2} \)
59 \( 1 + (-4.60 - 4.60i)T + 59iT^{2} \)
61 \( 1 + (0.211 - 0.211i)T - 61iT^{2} \)
67 \( 1 + (-7.17 - 7.17i)T + 67iT^{2} \)
71 \( 1 + 15.9iT - 71T^{2} \)
73 \( 1 - 10.5T + 73T^{2} \)
79 \( 1 - 4.53T + 79T^{2} \)
83 \( 1 + (4.56 + 4.56i)T + 83iT^{2} \)
89 \( 1 - 10.2iT - 89T^{2} \)
97 \( 1 + 6.78iT - 97T^{2} \)
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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

−9.955052994316326209785481671645, −9.204691481678801587420505816433, −8.441060457156219114314180055433, −7.63819355083411282365438604188, −6.47610662885528660596451133278, −5.71149149502178530855521568280, −5.14599455811843818437901803224, −3.80050383316832050118562942190, −2.68981774022378170025709023206, −0.29321295839041897090466999901, 2.08218207341279214095596819831, 2.73287302971876072192010413165, 3.97059087334160332103446627934, 5.06845493165260336080663605641, 6.08564664640118190451443736391, 6.88903108470608543429664465016, 8.111874241283700471458637453561, 9.460671958954763287620119025891, 9.795746761035296619858563604527, 10.64103598855697554392684222005

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