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Properties

Label 2-2880-48.11-c1-0-4
Degree $2$
Conductor $2880$
Sign $0.660 - 0.751i$
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 − 4.49·7-s + (−4.37 − 4.37i)11-s + (1.44 − 1.44i)13-s + 6.69i·17-s + (1.91 + 1.91i)19-s + 1.18i·23-s − 1.00i·25-s + (−2.16 − 2.16i)29-s + 4.28i·31-s + (−3.17 + 3.17i)35-s + (0.00767 + 0.00767i)37-s − 4.93·41-s + (−2.61 + 2.61i)43-s + 12.9·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s − 1.69·7-s + (−1.31 − 1.31i)11-s + (0.400 − 0.400i)13-s + 1.62i·17-s + (0.438 + 0.438i)19-s + 0.247i·23-s − 0.200i·25-s + (−0.401 − 0.401i)29-s + 0.769i·31-s + (−0.536 + 0.536i)35-s + (0.00126 + 0.00126i)37-s − 0.771·41-s + (−0.398 + 0.398i)43-s + 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.660 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.660 - 0.751i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.660 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9892913129\)
\(L(\frac12)\) \(\approx\) \(0.9892913129\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.49T + 7T^{2} \)
11 \( 1 + (4.37 + 4.37i)T + 11iT^{2} \)
13 \( 1 + (-1.44 + 1.44i)T - 13iT^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 + (-1.91 - 1.91i)T + 19iT^{2} \)
23 \( 1 - 1.18iT - 23T^{2} \)
29 \( 1 + (2.16 + 2.16i)T + 29iT^{2} \)
31 \( 1 - 4.28iT - 31T^{2} \)
37 \( 1 + (-0.00767 - 0.00767i)T + 37iT^{2} \)
41 \( 1 + 4.93T + 41T^{2} \)
43 \( 1 + (2.61 - 2.61i)T - 43iT^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + (-6.04 + 6.04i)T - 53iT^{2} \)
59 \( 1 + (-6.51 - 6.51i)T + 59iT^{2} \)
61 \( 1 + (-1.14 + 1.14i)T - 61iT^{2} \)
67 \( 1 + (-5.87 - 5.87i)T + 67iT^{2} \)
71 \( 1 + 8.93iT - 71T^{2} \)
73 \( 1 - 13.7iT - 73T^{2} \)
79 \( 1 - 4.10iT - 79T^{2} \)
83 \( 1 + (-4.52 + 4.52i)T - 83iT^{2} \)
89 \( 1 - 12.3T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−8.709416917932450916820900404509, −8.342640469460574912933663274595, −7.37835802356776595990756293409, −6.37564107532116005705241386140, −5.82637171981297818947676495367, −5.34141733652853420850559897541, −3.80629622135053910982818397104, −3.34974147318785425276789547485, −2.38974929985917829577794182117, −0.857669418765113896207063324428, 0.40289369721802316822334510921, 2.26685713176419726086444266219, 2.82620819541451237277489851984, 3.77966030832016235248273328809, 4.89524487390673081064566397295, 5.57749093238141465475448514369, 6.55976534402615877959416921220, 7.12004676737022019367914637178, 7.62942095609732981117695730237, 8.976042280244541489525430841766

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