Properties

Label 2-720-5.4-c3-0-35
Degree $2$
Conductor $720$
Sign $-0.447 + 0.894i$
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  + (5 − 10i)5-s + 26i·7-s − 28·11-s + 12i·13-s − 64i·17-s − 60·19-s − 58i·23-s + (−75 − 100i)25-s + 90·29-s + 128·31-s + (260 + 130i)35-s − 236i·37-s − 242·41-s − 362i·43-s − 226i·47-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + 1.40i·7-s − 0.767·11-s + 0.256i·13-s − 0.913i·17-s − 0.724·19-s − 0.525i·23-s + (−0.599 − 0.800i)25-s + 0.576·29-s + 0.741·31-s + (1.25 + 0.627i)35-s − 1.04i·37-s − 0.921·41-s − 1.28i·43-s − 0.701i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $-0.447 + 0.894i$
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: \((2,\ 720,\ (\ :3/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.109124206\)
\(L(\frac12)\) \(\approx\) \(1.109124206\)
\(L(\frac{5}{2})\) 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 \( 1 + (-5 + 10i)T \)
good7 \( 1 - 26iT - 343T^{2} \)
11 \( 1 + 28T + 1.33e3T^{2} \)
13 \( 1 - 12iT - 2.19e3T^{2} \)
17 \( 1 + 64iT - 4.91e3T^{2} \)
19 \( 1 + 60T + 6.85e3T^{2} \)
23 \( 1 + 58iT - 1.21e4T^{2} \)
29 \( 1 - 90T + 2.43e4T^{2} \)
31 \( 1 - 128T + 2.97e4T^{2} \)
37 \( 1 + 236iT - 5.06e4T^{2} \)
41 \( 1 + 242T + 6.89e4T^{2} \)
43 \( 1 + 362iT - 7.95e4T^{2} \)
47 \( 1 + 226iT - 1.03e5T^{2} \)
53 \( 1 - 108iT - 1.48e5T^{2} \)
59 \( 1 - 20T + 2.05e5T^{2} \)
61 \( 1 - 542T + 2.26e5T^{2} \)
67 \( 1 + 434iT - 3.00e5T^{2} \)
71 \( 1 + 1.12e3T + 3.57e5T^{2} \)
73 \( 1 - 632iT - 3.89e5T^{2} \)
79 \( 1 + 720T + 4.93e5T^{2} \)
83 \( 1 + 478iT - 5.71e5T^{2} \)
89 \( 1 + 490T + 7.04e5T^{2} \)
97 \( 1 + 1.45e3iT - 9.12e5T^{2} \)
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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

−9.646052268135818892784560341564, −8.695598428592748740083714589366, −8.437400582584077383016130373626, −7.06836026275220779523099613789, −5.92820828614388235569216177166, −5.29149880757721249256393405776, −4.43761069914329256552356823254, −2.77194529621760488438117723485, −1.95500920297652367921591402519, −0.30037764341700008923397956321, 1.32112351694799658375442404034, 2.71373281496499454479438303591, 3.72567478845097658737359769558, 4.77629299257984150591043866522, 6.04820468977411626267255619053, 6.77120246454101472728929849329, 7.64506717155168582949716136865, 8.380848014452515487858073964046, 9.816286128502080798848836558359, 10.34581645352371701977329874842

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