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Properties

Label 2-720-180.79-c0-0-0
Degree $2$
Conductor $720$
Sign $0.939 - 0.342i$
Analytic cond. $0.359326$
Root an. cond. $0.599438$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)15-s + 1.73i·21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 0.999i·27-s + (0.5 − 0.866i)29-s + 1.73·35-s + (−0.5 − 0.866i)41-s + 0.999·45-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + ⋯
L(s)  = 1  + (−0.866 + 0.5i)3-s + (0.5 + 0.866i)5-s + (0.866 − 1.5i)7-s + (0.499 − 0.866i)9-s + (−0.866 − 0.499i)15-s + 1.73i·21-s + (0.866 + 1.5i)23-s + (−0.499 + 0.866i)25-s + 0.999i·27-s + (0.5 − 0.866i)29-s + 1.73·35-s + (−0.5 − 0.866i)41-s + 0.999·45-s + (−0.866 + 1.5i)47-s + (−1 − 1.73i)49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.939 - 0.342i$
Analytic conductor: \(0.359326\)
Root analytic conductor: \(0.599438\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (79, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :0),\ 0.939 - 0.342i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.8736490765\)
\(L(\frac12)\) \(\approx\) \(0.8736490765\)
\(L(1)\) 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 \( 1 + (0.866 - 0.5i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
good7 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + (-0.866 - 1.5i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
43 \( 1 + (-0.5 - 0.866i)T^{2} \)
47 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + T + T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−10.75302865327000930474196888415, −10.05950253464592205343062015736, −9.290952778329490521865457765717, −7.77253046960430871630312415362, −7.12899306648116238315987085785, −6.26115072109761079650799173810, −5.22837144868878712016099572722, −4.31183165505352995031265383958, −3.32006138099948839028592176182, −1.43148663123145049578658092340, 1.45548708902427059096773852914, 2.51949460337761851653115025744, 4.69089718754580284850893298477, 5.16351289662131739329509394429, 5.98189069661683671444574195067, 6.87893465749138443788811077912, 8.288812447762607878337883643737, 8.608516016266899460232474759795, 9.697155589634592988573130281146, 10.73187450691698984967987388160

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