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Properties

Label 2-720-80.29-c1-0-48
Degree $2$
Conductor $720$
Sign $-0.991 + 0.133i$
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 + (−0.607 − 2.15i)5-s − 2.25·7-s + (0.0216 − 2.82i)8-s + (−0.785 + 3.06i)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 + (3.13 − 3.18i)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 + (−0.271 − 0.962i)5-s − 0.851·7-s + (0.00765 − 0.999i)8-s + (−0.248 + 0.968i)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 + (0.701 − 0.712i)20-s + (−0.684 + 0.181i)22-s − 0.921·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.991 + 0.133i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.991 + 0.133i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.991 + 0.133i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ -0.991 + 0.133i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0339242 - 0.505380i\)
\(L(\frac12)\) \(\approx\) \(0.0339242 - 0.505380i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.607 + 2.15i)T \)
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

−10.07587726674662997309066396612, −8.933032464766599238485128003985, −8.538565565012170340777713136343, −7.79584416393420092328022772383, −6.44417053695582257676395010409, −5.75157677760233545218768503515, −3.95870597232185566925175175034, −3.44779130041046353189026663753, −1.71635744411109027393117819192, −0.34642768277461777068880866669, 1.77717558982171961183253790060, 3.17286297266267930902933855923, 4.39075608703729202270987229990, 6.03323487027951450122739580471, 6.55282071300127555510727126015, 7.23123206739824351723668242147, 8.184954253807477477865467073571, 9.417635896316651962861678400822, 9.572199317234485773575557026745, 10.78254065481103008429397783425

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