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Properties

Label 2-720-80.69-c1-0-1
Degree $2$
Conductor $720$
Sign $-0.999 + 0.00910i$
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  + (−0.750 + 1.19i)2-s + (−0.874 − 1.79i)4-s + (−1.95 − 1.07i)5-s + 1.22·7-s + (2.81 + 0.302i)8-s + (2.76 − 1.54i)10-s + (−1.38 − 1.38i)11-s + (−2.12 − 2.12i)13-s + (−0.916 + 1.46i)14-s + (−2.47 + 3.14i)16-s + 6.00i·17-s + (−3.06 + 3.06i)19-s + (−0.225 + 4.46i)20-s + (2.69 − 0.619i)22-s + 2.90·23-s + ⋯
L(s)  = 1  + (−0.530 + 0.847i)2-s + (−0.437 − 0.899i)4-s + (−0.876 − 0.481i)5-s + 0.461·7-s + (0.994 + 0.106i)8-s + (0.873 − 0.487i)10-s + (−0.416 − 0.416i)11-s + (−0.588 − 0.588i)13-s + (−0.244 + 0.391i)14-s + (−0.618 + 0.786i)16-s + 1.45i·17-s + (−0.702 + 0.702i)19-s + (−0.0504 + 0.998i)20-s + (0.574 − 0.132i)22-s + 0.606·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.00910i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.00910i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.00910i$
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: \((2,\ 720,\ (\ :1/2),\ -0.999 + 0.00910i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00112032 - 0.246015i\)
\(L(\frac12)\) \(\approx\) \(0.00112032 - 0.246015i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.750 - 1.19i)T \)
3 \( 1 \)
5 \( 1 + (1.95 + 1.07i)T \)
good7 \( 1 - 1.22T + 7T^{2} \)
11 \( 1 + (1.38 + 1.38i)T + 11iT^{2} \)
13 \( 1 + (2.12 + 2.12i)T + 13iT^{2} \)
17 \( 1 - 6.00iT - 17T^{2} \)
19 \( 1 + (3.06 - 3.06i)T - 19iT^{2} \)
23 \( 1 - 2.90T + 23T^{2} \)
29 \( 1 + (3.18 - 3.18i)T - 29iT^{2} \)
31 \( 1 + 3.88T + 31T^{2} \)
37 \( 1 + (2.44 - 2.44i)T - 37iT^{2} \)
41 \( 1 + 2.38iT - 41T^{2} \)
43 \( 1 + (9.00 - 9.00i)T - 43iT^{2} \)
47 \( 1 - 0.586iT - 47T^{2} \)
53 \( 1 + (2.36 - 2.36i)T - 53iT^{2} \)
59 \( 1 + (8.43 + 8.43i)T + 59iT^{2} \)
61 \( 1 + (-9.98 + 9.98i)T - 61iT^{2} \)
67 \( 1 + (-3.82 - 3.82i)T + 67iT^{2} \)
71 \( 1 - 11.5iT - 71T^{2} \)
73 \( 1 + 1.31T + 73T^{2} \)
79 \( 1 + 12.5T + 79T^{2} \)
83 \( 1 + (2.91 + 2.91i)T + 83iT^{2} \)
89 \( 1 + 9.58iT - 89T^{2} \)
97 \( 1 - 9.45iT - 97T^{2} \)
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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

−10.74489195462219511451404169954, −9.908822286716062603109909928471, −8.754237809712751499731012825501, −8.192097344876116261957099719477, −7.65183557349107701184016064498, −6.55930658561699269293617459039, −5.48075640537055640807852673859, −4.72982170945939090815290658695, −3.57568445141680366772165699545, −1.55876522773403716447998670103, 0.15334433752313740062890187669, 2.10537761448167367636098072215, 3.10350343265854329615418665523, 4.31237143012845060744122453533, 5.03386109047321335529365294503, 7.01586808819424729226687188271, 7.35228272261275407959413794065, 8.382549058519306579572638829112, 9.195054328254861277321838297917, 10.06494807073951723945908144220

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