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Properties

Label 2-720-45.34-c1-0-14
Degree $2$
Conductor $720$
Sign $0.726 + 0.687i$
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  + (−0.866 − 1.5i)3-s + (−1.23 + 1.86i)5-s + (−0.866 − 0.5i)7-s + (−1.5 + 2.59i)9-s + (−1 + 1.73i)11-s + (5.19 − 3i)13-s + (3.86 + 0.232i)15-s − 2i·17-s + 6·19-s + 1.73i·21-s + (0.866 − 0.5i)23-s + (−1.96 − 4.59i)25-s + 5.19·27-s + (4.5 − 7.79i)29-s + (−1 − 1.73i)31-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)3-s + (−0.550 + 0.834i)5-s + (−0.327 − 0.188i)7-s + (−0.5 + 0.866i)9-s + (−0.301 + 0.522i)11-s + (1.44 − 0.832i)13-s + (0.998 + 0.0599i)15-s − 0.485i·17-s + 1.37·19-s + 0.377i·21-s + (0.180 − 0.104i)23-s + (−0.392 − 0.919i)25-s + 1.00·27-s + (0.835 − 1.44i)29-s + (−0.179 − 0.311i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 720 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.726 + 0.687i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(720\)    =    \(2^{4} \cdot 3^{2} \cdot 5\)
Sign: $0.726 + 0.687i$
Analytic conductor: \(5.74922\)
Root analytic conductor: \(2.39775\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{720} (529, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 720,\ (\ :1/2),\ 0.726 + 0.687i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.02892 - 0.409822i\)
\(L(\frac12)\) \(\approx\) \(1.02892 - 0.409822i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (0.866 + 1.5i)T \)
5 \( 1 + (1.23 - 1.86i)T \)
good7 \( 1 + (0.866 + 0.5i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (1 - 1.73i)T + (-5.5 - 9.52i)T^{2} \)
13 \( 1 + (-5.19 + 3i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 2iT - 17T^{2} \)
19 \( 1 - 6T + 19T^{2} \)
23 \( 1 + (-0.866 + 0.5i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.5 + 7.79i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1 + 1.73i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 - 2iT - 37T^{2} \)
41 \( 1 + (-5.5 - 9.52i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.46 - 2i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.06 + 3.5i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 - 53T^{2} \)
59 \( 1 + (-2 - 3.46i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + (-3.5 + 6.06i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-9.52 + 5.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 - 4iT - 73T^{2} \)
79 \( 1 + (-6 + 10.3i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-9.52 - 5.5i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 + (-6.92 - 4i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.46783840595048207621371511240, −9.635473224972143342531281860867, −8.106679360268048411670211141711, −7.76964885821919335219553680533, −6.72704430559583901367079952899, −6.12226188138329228722082746667, −5.00075074395910026035700728142, −3.57034577299646885255250614490, −2.58228387552900727560411759714, −0.824705322894837763088374231607, 1.06429312710593128347626894928, 3.30835354477426123233332090534, 4.00817826337069251079278671397, 5.12771741220225867000129384439, 5.82266819311640273788248539442, 6.88199764952872004576816325245, 8.211974599608282287201586437495, 8.938950287996966059231709305992, 9.480929547280264828305594998512, 10.72249341754975827139252861767

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