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Properties

Label 2-2880-45.14-c0-0-3
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: $\chi_{2880} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.984 - 0.173i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.961358181\)
\(L(\frac12)\) \(\approx\) \(1.961358181\)
\(L(1)\) 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

−9.154388512506697609785969649069, −8.176070585181918332949646355660, −7.55642678759081206090927163604, −6.66439053003908274560618539892, −6.13026734662188961093523387493, −5.20752007966247434871827305547, −4.00431419594676686278454369245, −3.18229935305726086906130503203, −2.41245072069825202993507061223, −1.53439041674574489341111174240, 1.31646237236159733695657026765, 2.42862251741503901800628226426, 3.13027967074366631860804573644, 4.33498417993268552585678512277, 4.78910190551763356074877052522, 5.99952391547570540233086379811, 6.61845126291664423633590312175, 7.53641863627215697060135151058, 8.481442252099955315905773241491, 8.805610825052958731109797439918

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