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Properties

Label 2-2880-16.13-c1-0-18
Degree $2$
Conductor $2880$
Sign $0.944 - 0.329i$
Analytic cond. $22.9969$
Root an. cond. $4.79550$
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  + (0.707 − 0.707i)5-s + 2.69i·7-s + (2.72 − 2.72i)11-s + (1.82 + 1.82i)13-s + 7.33·17-s + (−3.62 − 3.62i)19-s + 8.95i·23-s − 1.00i·25-s + (−2.84 − 2.84i)29-s + 3.37·31-s + (1.90 + 1.90i)35-s + (−0.190 + 0.190i)37-s − 7.67i·41-s + (−7.98 + 7.98i)43-s − 1.31·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s + 1.01i·7-s + (0.822 − 0.822i)11-s + (0.506 + 0.506i)13-s + 1.77·17-s + (−0.831 − 0.831i)19-s + 1.86i·23-s − 0.200i·25-s + (−0.527 − 0.527i)29-s + 0.607·31-s + (0.322 + 0.322i)35-s + (−0.0312 + 0.0312i)37-s − 1.19i·41-s + (−1.21 + 1.21i)43-s − 0.191·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.944 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.944 - 0.329i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.944 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.197129717\)
\(L(\frac12)\) \(\approx\) \(2.197129717\)
\(L(\frac{3}{2})\) 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 \( 1 \)
5 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - 2.69iT - 7T^{2} \)
11 \( 1 + (-2.72 + 2.72i)T - 11iT^{2} \)
13 \( 1 + (-1.82 - 1.82i)T + 13iT^{2} \)
17 \( 1 - 7.33T + 17T^{2} \)
19 \( 1 + (3.62 + 3.62i)T + 19iT^{2} \)
23 \( 1 - 8.95iT - 23T^{2} \)
29 \( 1 + (2.84 + 2.84i)T + 29iT^{2} \)
31 \( 1 - 3.37T + 31T^{2} \)
37 \( 1 + (0.190 - 0.190i)T - 37iT^{2} \)
41 \( 1 + 7.67iT - 41T^{2} \)
43 \( 1 + (7.98 - 7.98i)T - 43iT^{2} \)
47 \( 1 + 1.31T + 47T^{2} \)
53 \( 1 + (-6.71 + 6.71i)T - 53iT^{2} \)
59 \( 1 + (-1.01 + 1.01i)T - 59iT^{2} \)
61 \( 1 + (-2.38 - 2.38i)T + 61iT^{2} \)
67 \( 1 + (-7.22 - 7.22i)T + 67iT^{2} \)
71 \( 1 - 2.28iT - 71T^{2} \)
73 \( 1 - 1.31iT - 73T^{2} \)
79 \( 1 - 2.59T + 79T^{2} \)
83 \( 1 + (5.36 + 5.36i)T + 83iT^{2} \)
89 \( 1 - 14.8iT - 89T^{2} \)
97 \( 1 - 0.694T + 97T^{2} \)
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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

−8.749778382191619567755340402834, −8.320634469505672097170786801540, −7.29992386817396078448100673523, −6.35477537533922560622027026943, −5.72611761209372586729023018015, −5.18046305939104956392354285975, −3.93662066075710820471225092449, −3.22955917397162471427767344394, −2.05663706408145125479831914717, −1.05689089589911653026583276405, 0.890330992478198489518394525372, 1.90221869046872592672937470950, 3.20727283564282921345974594416, 3.93021854967571539894197716481, 4.73425657409177823436619167128, 5.79229786022171564193432138680, 6.52510511648244245034421352559, 7.15388636643458096738220732643, 7.973842441564109421172020260662, 8.621045951187322899287866869279

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