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Properties

Label 2-720-80.69-c1-0-48
Degree $2$
Conductor $720$
Sign $0.404 + 0.914i$
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  + (1.22 − 0.710i)2-s + (0.991 − 1.73i)4-s + (2.15 − 0.607i)5-s + 2.25·7-s + (−0.0216 − 2.82i)8-s + (2.20 − 2.27i)10-s + (1.66 + 1.66i)11-s + (−4.76 − 4.76i)13-s + (2.75 − 1.60i)14-s + (−2.03 − 3.44i)16-s + 6.99i·17-s + (−2.66 + 2.66i)19-s + (1.07 − 4.34i)20-s + (3.21 + 0.851i)22-s + 4.41·23-s + ⋯
L(s)  = 1  + (0.864 − 0.502i)2-s + (0.495 − 0.868i)4-s + (0.962 − 0.271i)5-s + 0.851·7-s + (−0.00765 − 0.999i)8-s + (0.695 − 0.718i)10-s + (0.500 + 0.500i)11-s + (−1.32 − 1.32i)13-s + (0.736 − 0.427i)14-s + (−0.508 − 0.860i)16-s + 1.69i·17-s + (−0.611 + 0.611i)19-s + (0.240 − 0.970i)20-s + (0.684 + 0.181i)22-s + 0.921·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.404 + 0.914i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.404 + 0.914i$
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: \((2,\ 720,\ (\ :1/2),\ 0.404 + 0.914i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.55835 - 1.66616i\)
\(L(\frac12)\) \(\approx\) \(2.55835 - 1.66616i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.22 + 0.710i)T \)
3 \( 1 \)
5 \( 1 + (-2.15 + 0.607i)T \)
good7 \( 1 - 2.25T + 7T^{2} \)
11 \( 1 + (-1.66 - 1.66i)T + 11iT^{2} \)
13 \( 1 + (4.76 + 4.76i)T + 13iT^{2} \)
17 \( 1 - 6.99iT - 17T^{2} \)
19 \( 1 + (2.66 - 2.66i)T - 19iT^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + (2.59 - 2.59i)T - 29iT^{2} \)
31 \( 1 + 3.93T + 31T^{2} \)
37 \( 1 + (2.01 - 2.01i)T - 37iT^{2} \)
41 \( 1 - 4.50iT - 41T^{2} \)
43 \( 1 + (-7.14 + 7.14i)T - 43iT^{2} \)
47 \( 1 + 10.1iT - 47T^{2} \)
53 \( 1 + (0.649 - 0.649i)T - 53iT^{2} \)
59 \( 1 + (5.64 + 5.64i)T + 59iT^{2} \)
61 \( 1 + (5.00 - 5.00i)T - 61iT^{2} \)
67 \( 1 + (-4.95 - 4.95i)T + 67iT^{2} \)
71 \( 1 - 2.33iT - 71T^{2} \)
73 \( 1 + 2.18T + 73T^{2} \)
79 \( 1 - 6.38T + 79T^{2} \)
83 \( 1 + (-5.25 - 5.25i)T + 83iT^{2} \)
89 \( 1 - 15.7iT - 89T^{2} \)
97 \( 1 - 4.61iT - 97T^{2} \)
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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

−10.43069391161099554913004720846, −9.703794696522157685189022837445, −8.646749115044785725827178682222, −7.52283106569048529785029709444, −6.44429894400693965306381364600, −5.47039859187407342056585656539, −4.92593809170610280336830492729, −3.76084202934486792271777562694, −2.34878901127972998385603139016, −1.47251254231717232344313099615, 1.98213661759937223526373799476, 2.91324550960202578681419468022, 4.51707929467299751620110070177, 5.02067987936049368251440754513, 6.10889675093048117092945571389, 7.00190585938981925583072018778, 7.56633463509110764869839301735, 9.056356641716828977637213912460, 9.369669555053132312912995105756, 10.95106223984424311446923894241

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