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Properties

Label 2-2880-120.53-c1-0-36
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: $\chi_{2880} (1313, \cdot )$
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

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

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