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Properties

Label 2-2880-48.11-c1-0-17
Degree $2$
Conductor $2880$
Sign $0.669 + 0.742i$
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 − 1.44·7-s + (1.89 + 1.89i)11-s + (0.905 − 0.905i)13-s − 6.42i·17-s + (2.02 + 2.02i)19-s + 4.25i·23-s − 1.00i·25-s + (−2.71 − 2.71i)29-s − 4.19i·31-s + (−1.02 + 1.02i)35-s + (0.0486 + 0.0486i)37-s + 8.11·41-s + (3.17 − 3.17i)43-s + 2.47·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s − 0.546·7-s + (0.571 + 0.571i)11-s + (0.251 − 0.251i)13-s − 1.55i·17-s + (0.464 + 0.464i)19-s + 0.887i·23-s − 0.200i·25-s + (−0.504 − 0.504i)29-s − 0.752i·31-s + (−0.172 + 0.172i)35-s + (0.00800 + 0.00800i)37-s + 1.26·41-s + (0.484 − 0.484i)43-s + 0.361·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.669 + 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.669 + 0.742i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.669 + 0.742i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.803386588\)
\(L(\frac12)\) \(\approx\) \(1.803386588\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.44T + 7T^{2} \)
11 \( 1 + (-1.89 - 1.89i)T + 11iT^{2} \)
13 \( 1 + (-0.905 + 0.905i)T - 13iT^{2} \)
17 \( 1 + 6.42iT - 17T^{2} \)
19 \( 1 + (-2.02 - 2.02i)T + 19iT^{2} \)
23 \( 1 - 4.25iT - 23T^{2} \)
29 \( 1 + (2.71 + 2.71i)T + 29iT^{2} \)
31 \( 1 + 4.19iT - 31T^{2} \)
37 \( 1 + (-0.0486 - 0.0486i)T + 37iT^{2} \)
41 \( 1 - 8.11T + 41T^{2} \)
43 \( 1 + (-3.17 + 3.17i)T - 43iT^{2} \)
47 \( 1 - 2.47T + 47T^{2} \)
53 \( 1 + (1.67 - 1.67i)T - 53iT^{2} \)
59 \( 1 + (-4.69 - 4.69i)T + 59iT^{2} \)
61 \( 1 + (-2.69 + 2.69i)T - 61iT^{2} \)
67 \( 1 + (8.13 + 8.13i)T + 67iT^{2} \)
71 \( 1 - 13.8iT - 71T^{2} \)
73 \( 1 + 5.23iT - 73T^{2} \)
79 \( 1 + 6.59iT - 79T^{2} \)
83 \( 1 + (-12.8 + 12.8i)T - 83iT^{2} \)
89 \( 1 - 17.0T + 89T^{2} \)
97 \( 1 - 12.8T + 97T^{2} \)
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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

−8.921220278066596817232717984126, −7.69862946664836584993822548006, −7.32544885258958897422330670602, −6.29349099434901630808731227843, −5.66682846954557440123161339369, −4.80160582895826783305150085556, −3.90155071279619789031433748352, −2.98752718013369601388990964636, −1.92651479160664770344014641151, −0.67890257061039864954812749991, 1.06698224616326562681354242495, 2.25509386012447364277960669497, 3.32022263732834561238545458595, 3.96418575491388118621843757248, 5.04525783262299197068503530232, 6.12777276208686481513942267844, 6.36646066348314610183358569464, 7.28139638627332606523392641619, 8.213566202628019294692256439385, 8.944921519797420487048745178627

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