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Properties

Label 2-2880-45.29-c0-0-2
Degree $2$
Conductor $2880$
Sign $0.984 + 0.173i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.866 − 0.5i)5-s + (0.448 + 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)15-s + (−0.366 − 0.366i)21-s + (0.965 + 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (1.5 + 0.866i)29-s + 0.517·35-s + (−0.866 + 0.5i)41-s + (−1.22 − 0.707i)43-s + 45-s + (0.258 − 0.448i)47-s + ⋯
L(s)  = 1  + (−0.965 − 0.258i)3-s + (0.866 − 0.5i)5-s + (0.448 + 0.258i)7-s + (0.866 + 0.499i)9-s + (−0.965 + 0.258i)15-s + (−0.366 − 0.366i)21-s + (0.965 + 1.67i)23-s + (0.499 − 0.866i)25-s + (−0.707 − 0.707i)27-s + (1.5 + 0.866i)29-s + 0.517·35-s + (−0.866 + 0.5i)41-s + (−1.22 − 0.707i)43-s + 45-s + (0.258 − 0.448i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.984 + 0.173i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2369, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.984 + 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.164160884\)
\(L(\frac12)\) \(\approx\) \(1.164160884\)
\(L(1)\) 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 \( 1 + (0.965 + 0.258i)T \)
5 \( 1 + (-0.866 + 0.5i)T \)
good7 \( 1 + (-0.448 - 0.258i)T + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (0.5 - 0.866i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + (-0.965 - 1.67i)T + (-0.5 + 0.866i)T^{2} \)
29 \( 1 + (-1.5 - 0.866i)T + (0.5 + 0.866i)T^{2} \)
31 \( 1 + (-0.5 + 0.866i)T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
43 \( 1 + (1.22 + 0.707i)T + (0.5 + 0.866i)T^{2} \)
47 \( 1 + (-0.258 + 0.448i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + T^{2} \)
59 \( 1 + (0.5 - 0.866i)T^{2} \)
61 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-1.67 + 0.965i)T + (0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 + (-0.5 - 0.866i)T^{2} \)
83 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
89 \( 1 + 1.73iT - T^{2} \)
97 \( 1 + (0.5 + 0.866i)T^{2} \)
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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

−8.893988660036206571711637735507, −8.270896077190220514796266015091, −7.21254343964388629197875363367, −6.63015788367921801464467409227, −5.72867220046376992307803364034, −5.16194781678088058851502440198, −4.64083547859337463341824743118, −3.26973733809854926431712769945, −1.94449489342894784617055956240, −1.18008220266931462806603336171, 1.06329230811695786670522117488, 2.28501861680025308415867257342, 3.36762222303188519336907381593, 4.64978765935601623236265620424, 4.95569741580971307860140900359, 6.10291543468735316858849217670, 6.49909979603765181194864840797, 7.22006018194733515533881775381, 8.258930832664220660581996825814, 9.081380870527033252235969878148

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