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Properties

Label 2-2880-120.77-c1-0-24
Degree $2$
Conductor $2880$
Sign $0.812 - 0.583i$
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  + (2.22 + 0.224i)5-s + (1.41 + 1.41i)7-s + 3.46·11-s + (0.317 + 0.317i)13-s + (−2.04 + 2.04i)17-s + 2.89·19-s + (−0.449 − 0.449i)23-s + (4.89 + i)25-s + 3.55i·29-s + 4.09·31-s + (2.82 + 3.46i)35-s + (−5.97 + 5.97i)37-s + 1.41i·41-s + (−2.89 − 2.89i)43-s + (6.44 − 6.44i)47-s + ⋯
L(s)  = 1  + (0.994 + 0.100i)5-s + (0.534 + 0.534i)7-s + 1.04·11-s + (0.0881 + 0.0881i)13-s + (−0.497 + 0.497i)17-s + 0.665·19-s + (−0.0937 − 0.0937i)23-s + (0.979 + 0.200i)25-s + 0.659i·29-s + 0.736·31-s + (0.478 + 0.585i)35-s + (−0.982 + 0.982i)37-s + 0.220i·41-s + (−0.442 − 0.442i)43-s + (0.940 − 0.940i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.812 - 0.583i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.812 - 0.583i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.621766878\)
\(L(\frac12)\) \(\approx\) \(2.621766878\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 0.224i)T \)
good7 \( 1 + (-1.41 - 1.41i)T + 7iT^{2} \)
11 \( 1 - 3.46T + 11T^{2} \)
13 \( 1 + (-0.317 - 0.317i)T + 13iT^{2} \)
17 \( 1 + (2.04 - 2.04i)T - 17iT^{2} \)
19 \( 1 - 2.89T + 19T^{2} \)
23 \( 1 + (0.449 + 0.449i)T + 23iT^{2} \)
29 \( 1 - 3.55iT - 29T^{2} \)
31 \( 1 - 4.09T + 31T^{2} \)
37 \( 1 + (5.97 - 5.97i)T - 37iT^{2} \)
41 \( 1 - 1.41iT - 41T^{2} \)
43 \( 1 + (2.89 + 2.89i)T + 43iT^{2} \)
47 \( 1 + (-6.44 + 6.44i)T - 47iT^{2} \)
53 \( 1 + (-2.89 + 2.89i)T - 53iT^{2} \)
59 \( 1 + 6.29iT - 59T^{2} \)
61 \( 1 - 13.2iT - 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 + 12.8iT - 71T^{2} \)
73 \( 1 + (7.89 - 7.89i)T - 73iT^{2} \)
79 \( 1 - 14.1iT - 79T^{2} \)
83 \( 1 + (9.12 - 9.12i)T - 83iT^{2} \)
89 \( 1 - 4.24T + 89T^{2} \)
97 \( 1 + (7.89 + 7.89i)T + 97iT^{2} \)
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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

−8.688480437677190891712969752663, −8.472559344657717251585933971164, −7.09729531126098561597550778060, −6.61265534970270822943862594124, −5.73598393691726267537481577466, −5.12829040986462732427879555127, −4.16190693233324044771207391535, −3.10515102986933239931737087575, −2.05114316969647092964642306116, −1.28477929478183782357809104864, 0.949332279526225726823341097661, 1.85274027569802619524647285969, 2.92811472290469719801705698402, 4.06758791824844468144911282763, 4.78292948063906071672413614580, 5.68160670286334184995504406363, 6.39357232581614722375245552485, 7.14746335771806154186197210570, 7.896560330303643188579892323972, 8.960851038670724411914830092895

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