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Properties

Label 2-720-80.69-c1-0-42
Degree $2$
Conductor $720$
Sign $0.326 + 0.945i$
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  + (1.06 − 0.933i)2-s + (0.256 − 1.98i)4-s + (1.86 + 1.24i)5-s + 1.58·7-s + (−1.57 − 2.34i)8-s + (3.13 − 0.418i)10-s + (−3.92 − 3.92i)11-s + (3.10 + 3.10i)13-s + (1.68 − 1.48i)14-s + (−3.86 − 1.01i)16-s − 1.48i·17-s + (4.94 − 4.94i)19-s + (2.93 − 3.37i)20-s + (−7.82 − 0.504i)22-s + 6.61·23-s + ⋯
L(s)  = 1  + (0.751 − 0.660i)2-s + (0.128 − 0.991i)4-s + (0.831 + 0.554i)5-s + 0.600·7-s + (−0.558 − 0.829i)8-s + (0.991 − 0.132i)10-s + (−1.18 − 1.18i)11-s + (0.861 + 0.861i)13-s + (0.451 − 0.396i)14-s + (−0.967 − 0.254i)16-s − 0.359i·17-s + (1.13 − 1.13i)19-s + (0.657 − 0.753i)20-s + (−1.66 − 0.107i)22-s + 1.38·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.326 + 0.945i$
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: \((2,\ 720,\ (\ :1/2),\ 0.326 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.17736 - 1.55173i\)
\(L(\frac12)\) \(\approx\) \(2.17736 - 1.55173i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.06 + 0.933i)T \)
3 \( 1 \)
5 \( 1 + (-1.86 - 1.24i)T \)
good7 \( 1 - 1.58T + 7T^{2} \)
11 \( 1 + (3.92 + 3.92i)T + 11iT^{2} \)
13 \( 1 + (-3.10 - 3.10i)T + 13iT^{2} \)
17 \( 1 + 1.48iT - 17T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 19iT^{2} \)
23 \( 1 - 6.61T + 23T^{2} \)
29 \( 1 + (4.42 - 4.42i)T - 29iT^{2} \)
31 \( 1 + 1.50T + 31T^{2} \)
37 \( 1 + (2.14 - 2.14i)T - 37iT^{2} \)
41 \( 1 + 6.84iT - 41T^{2} \)
43 \( 1 + (-0.322 + 0.322i)T - 43iT^{2} \)
47 \( 1 - 13.3iT - 47T^{2} \)
53 \( 1 + (-0.931 + 0.931i)T - 53iT^{2} \)
59 \( 1 + (1.14 + 1.14i)T + 59iT^{2} \)
61 \( 1 + (-2.67 + 2.67i)T - 61iT^{2} \)
67 \( 1 + (5.43 + 5.43i)T + 67iT^{2} \)
71 \( 1 - 2.26iT - 71T^{2} \)
73 \( 1 + 5.27T + 73T^{2} \)
79 \( 1 + 6.52T + 79T^{2} \)
83 \( 1 + (-0.973 - 0.973i)T + 83iT^{2} \)
89 \( 1 - 6.83iT - 89T^{2} \)
97 \( 1 - 15.8iT - 97T^{2} \)
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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

−10.76561381432710069904158843122, −9.427465143619054467376571642409, −8.885392206337325581615488536741, −7.41236963757658625654614079764, −6.49551047089404081953880655417, −5.47898339932503334944412116492, −4.96412357291257182966751011235, −3.39978627450740499622930319796, −2.65160270702057529155919705223, −1.29273548445432039625826502642, 1.77159768536328968402056904550, 3.10108751803041037021516224819, 4.44480906474682982933123738208, 5.38489880312812810217240528853, 5.73416300188648983779480291279, 7.10387811663834793738905250910, 7.898477552392698818870181553296, 8.581713741234396152133773283228, 9.717486159874536678493140685989, 10.54197012093378256988619074383

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