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Properties

Label 2-2880-16.5-c1-0-38
Degree $2$
Conductor $2880$
Sign $-0.992 + 0.118i$
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 − 1.47i·7-s + (−1.91 − 1.91i)11-s + (4.06 − 4.06i)13-s − 1.32·17-s + (−3.46 + 3.46i)19-s − 7.30i·23-s + 1.00i·25-s + (−4.04 + 4.04i)29-s − 0.0828·31-s + (−1.04 + 1.04i)35-s + (4.52 + 4.52i)37-s + 0.696i·41-s + (−6.70 − 6.70i)43-s − 8.68·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 0.558i·7-s + (−0.578 − 0.578i)11-s + (1.12 − 1.12i)13-s − 0.321·17-s + (−0.795 + 0.795i)19-s − 1.52i·23-s + 0.200i·25-s + (−0.750 + 0.750i)29-s − 0.0148·31-s + (−0.176 + 0.176i)35-s + (0.744 + 0.744i)37-s + 0.108i·41-s + (−1.02 − 1.02i)43-s − 1.26·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.992 + 0.118i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.992 + 0.118i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.992 + 0.118i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6563854479\)
\(L(\frac12)\) \(\approx\) \(0.6563854479\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.47iT - 7T^{2} \)
11 \( 1 + (1.91 + 1.91i)T + 11iT^{2} \)
13 \( 1 + (-4.06 + 4.06i)T - 13iT^{2} \)
17 \( 1 + 1.32T + 17T^{2} \)
19 \( 1 + (3.46 - 3.46i)T - 19iT^{2} \)
23 \( 1 + 7.30iT - 23T^{2} \)
29 \( 1 + (4.04 - 4.04i)T - 29iT^{2} \)
31 \( 1 + 0.0828T + 31T^{2} \)
37 \( 1 + (-4.52 - 4.52i)T + 37iT^{2} \)
41 \( 1 - 0.696iT - 41T^{2} \)
43 \( 1 + (6.70 + 6.70i)T + 43iT^{2} \)
47 \( 1 + 8.68T + 47T^{2} \)
53 \( 1 + (-4.80 - 4.80i)T + 53iT^{2} \)
59 \( 1 + (0.484 + 0.484i)T + 59iT^{2} \)
61 \( 1 + (6.60 - 6.60i)T - 61iT^{2} \)
67 \( 1 + (0.600 - 0.600i)T - 67iT^{2} \)
71 \( 1 - 7.46iT - 71T^{2} \)
73 \( 1 + 12.5iT - 73T^{2} \)
79 \( 1 - 2.25T + 79T^{2} \)
83 \( 1 + (8.28 - 8.28i)T - 83iT^{2} \)
89 \( 1 + 6.83iT - 89T^{2} \)
97 \( 1 - 6.90T + 97T^{2} \)
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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

−8.352591598293019396377710447626, −7.892607561618020288525432617937, −6.87389563025648608611419001450, −6.08794073833138548944916154836, −5.36933107943765032696974572887, −4.38336981521466798030018952364, −3.64262485973599735811775794473, −2.76374096184783358413374474083, −1.36074971184855534074623665611, −0.20931511702289925779897877440, 1.65811782445556436724076141045, 2.51073228396110618734314386414, 3.63399137551409222192480830563, 4.36777933158960589637561281517, 5.29194231993351939810407132605, 6.19434426848540121040729898905, 6.81641324906920510426946056494, 7.65947951825606254562961869647, 8.370343841358842103212151534561, 9.189362099172826217041713514113

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