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Properties

Label 2-2880-5.4-c1-0-46
Degree $2$
Conductor $2880$
Sign $-0.447 + 0.894i$
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  + (1 − 2i)5-s + 4i·13-s − 8i·17-s + (−3 − 4i)25-s − 10·29-s − 12i·37-s + 10·41-s + 7·49-s + 4i·53-s − 10·61-s + (8 + 4i)65-s − 16i·73-s + (−16 − 8i)85-s − 10·89-s − 8i·97-s + ⋯
L(s)  = 1  + (0.447 − 0.894i)5-s + 1.10i·13-s − 1.94i·17-s + (−0.600 − 0.800i)25-s − 1.85·29-s − 1.97i·37-s + 1.56·41-s + 49-s + 0.549i·53-s − 1.28·61-s + (0.992 + 0.496i)65-s − 1.87i·73-s + (−1.73 − 0.867i)85-s − 1.05·89-s − 0.812i·97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.447 + 0.894i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.447 + 0.894i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.431706806\)
\(L(\frac12)\) \(\approx\) \(1.431706806\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1 + 2i)T \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 + 8iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 + 10T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + 12iT - 37T^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 4iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 10T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 16iT - 73T^{2} \)
79 \( 1 + 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 + 10T + 89T^{2} \)
97 \( 1 + 8iT - 97T^{2} \)
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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

−8.843930245042817580366110826752, −7.57528557064141829846046065116, −7.23977407670089953404899970467, −6.07654827596394856043136690629, −5.45548354361408309909675659107, −4.62227008353658746517349341601, −3.93645984749375776154401942758, −2.61014959165013437449921065583, −1.73915793170354511475675263602, −0.44478904099776668615566349508, 1.42488959138947030138475184456, 2.46928385096918822876953736359, 3.38710099032025461294869529499, 4.12352554763270927032121393482, 5.43672455959816009202069942142, 5.94202489716515719519305952189, 6.65663653968476003622678429574, 7.59812668664369178690132370819, 8.121301343405760959183482689883, 9.039523538675511724137548248194

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