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Properties

Label 2-2880-48.11-c1-0-22
Degree $2$
Conductor $2880$
Sign $-0.476 + 0.879i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.476 + 0.879i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.476 + 0.879i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.476 + 0.879i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6090700704\)
\(L(\frac12)\) \(\approx\) \(0.6090700704\)
\(L(\frac{3}{2})\) 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

−8.268587554948845337662032956272, −8.037784122086218194670170909279, −6.91956065671862085376329931368, −6.24144597642477095286992857630, −5.60222306908195620961402911611, −4.53261103254967635613048227544, −3.56509235825869998425655207898, −2.99755871817167416125802169798, −1.72880341358358565881487707608, −0.20599604457263724260159797710, 1.18997006931509862513204426146, 2.59010344404440944669693853788, 3.31319648113533700533930361809, 4.44123411077673097218485527263, 5.07325540836975173606260829283, 5.90133183684393474891271059962, 7.05022468003292719198553780990, 7.28965221020184268124786066472, 8.320003697491441921005998277224, 9.035453590379403306415739069712

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