Properties

Label 2-720-5.4-c3-0-13
Degree $2$
Conductor $720$
Sign $0.993 - 0.116i$
Analytic cond. $42.4813$
Root an. cond. $6.51777$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−11.1 + 1.29i)5-s − 16.2i·7-s − 40.2·11-s + 19.7i·13-s + 83.0i·17-s − 48.8·19-s + 1.61i·23-s + (121. − 28.8i)25-s − 24.5·29-s + 12.4·31-s + (21.0 + 180i)35-s − 325. i·37-s + 242.·41-s − 367. i·43-s + 204. i·47-s + ⋯
L(s)  = 1  + (−0.993 + 0.116i)5-s − 0.875i·7-s − 1.10·11-s + 0.422i·13-s + 1.18i·17-s − 0.589·19-s + 0.0146i·23-s + (0.973 − 0.230i)25-s − 0.157·29-s + 0.0719·31-s + (0.101 + 0.869i)35-s − 1.44i·37-s + 0.923·41-s − 1.30i·43-s + 0.634i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.993 - 0.116i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.993 - 0.116i$
Analytic conductor: \(42.4813\)
Root analytic conductor: \(6.51777\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :3/2),\ 0.993 - 0.116i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.152497587\)
\(L(\frac12)\) \(\approx\) \(1.152497587\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 \)
5 \( 1 + (11.1 - 1.29i)T \)
good7 \( 1 + 16.2iT - 343T^{2} \)
11 \( 1 + 40.2T + 1.33e3T^{2} \)
13 \( 1 - 19.7iT - 2.19e3T^{2} \)
17 \( 1 - 83.0iT - 4.91e3T^{2} \)
19 \( 1 + 48.8T + 6.85e3T^{2} \)
23 \( 1 - 1.61iT - 1.21e4T^{2} \)
29 \( 1 + 24.5T + 2.43e4T^{2} \)
31 \( 1 - 12.4T + 2.97e4T^{2} \)
37 \( 1 + 325. iT - 5.06e4T^{2} \)
41 \( 1 - 242.T + 6.89e4T^{2} \)
43 \( 1 + 367. iT - 7.95e4T^{2} \)
47 \( 1 - 204. iT - 1.03e5T^{2} \)
53 \( 1 - 61.5iT - 1.48e5T^{2} \)
59 \( 1 - 112.T + 2.05e5T^{2} \)
61 \( 1 - 477.T + 2.26e5T^{2} \)
67 \( 1 - 558. iT - 3.00e5T^{2} \)
71 \( 1 - 558.T + 3.57e5T^{2} \)
73 \( 1 - 1.01e3iT - 3.89e5T^{2} \)
79 \( 1 - 1.15e3T + 4.93e5T^{2} \)
83 \( 1 + 1.15e3iT - 5.71e5T^{2} \)
89 \( 1 - 96.9T + 7.04e5T^{2} \)
97 \( 1 - 1.15e3iT - 9.12e5T^{2} \)
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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

−10.40841846522510466958876703423, −9.055211770357389017011995991807, −8.137392370390727704203225957035, −7.51954984049168046884325541718, −6.69424388271704429931121888263, −5.49485973972768383863281247451, −4.28093030713821069968264573978, −3.71033914881123060810941537211, −2.29593398534495789170469447916, −0.64100420281826216527093165951, 0.55258587749737921052581322302, 2.41318541054096188250555171794, 3.28430106088384710439753590585, 4.65462759898636209791835492958, 5.32839299599788793662937447165, 6.50241972965961002212713433313, 7.60585152766906665037572370727, 8.183962045386672191228387611063, 9.028559137295051060005459865619, 10.00088695171240247143042115113

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