Properties

Label 2-2880-24.11-c1-0-23
Degree $2$
Conductor $2880$
Sign $0.169 + 0.985i$
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  − 5-s − 3.16i·7-s − 5.16i·11-s + 3.05i·13-s + 7.30i·17-s + 7.30·19-s + 25-s + 4.32·29-s + 8.32i·31-s + 3.16i·35-s − 11.5i·37-s − 10.3i·41-s + 8.48·43-s − 1.18·47-s − 3.00·49-s + ⋯
L(s)  = 1  − 0.447·5-s − 1.19i·7-s − 1.55i·11-s + 0.848i·13-s + 1.77i·17-s + 1.67·19-s + 0.200·25-s + 0.803·29-s + 1.49i·31-s + 0.534i·35-s − 1.89i·37-s − 1.61i·41-s + 1.29·43-s − 0.172·47-s − 0.428·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.169 + 0.985i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2591, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.543748058\)
\(L(\frac12)\) \(\approx\) \(1.543748058\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good7 \( 1 + 3.16iT - 7T^{2} \)
11 \( 1 + 5.16iT - 11T^{2} \)
13 \( 1 - 3.05iT - 13T^{2} \)
17 \( 1 - 7.30iT - 17T^{2} \)
19 \( 1 - 7.30T + 19T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 - 4.32T + 29T^{2} \)
31 \( 1 - 8.32iT - 31T^{2} \)
37 \( 1 + 11.5iT - 37T^{2} \)
41 \( 1 + 10.3iT - 41T^{2} \)
43 \( 1 - 8.48T + 43T^{2} \)
47 \( 1 + 1.18T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 + 6.83iT - 59T^{2} \)
61 \( 1 + 8.48iT - 61T^{2} \)
67 \( 1 + 6.11T + 67T^{2} \)
71 \( 1 + 6.11T + 71T^{2} \)
73 \( 1 + 2T + 73T^{2} \)
79 \( 1 + 12.3iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 - 1.87iT - 89T^{2} \)
97 \( 1 - 12.3T + 97T^{2} \)
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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

−8.630798485512463431910530183249, −7.76723572184700664923757151062, −7.22286326188322200583161368566, −6.33381411492440253711795950062, −5.61859042132559132323207378934, −4.53783652431009991920181676307, −3.70116068799211427084607979300, −3.25412659549503768301830785058, −1.61491148899301569721629135857, −0.58184783937199424772887929117, 1.12515719127505089303640734221, 2.62610731972336804167084421317, 2.97288406034477227594862936209, 4.45856500575840239529729435556, 5.01431351116814259464399527637, 5.76167172680446302020463668838, 6.77209114551827279231746062463, 7.58881894995609991306677266344, 7.977861141001479577653829033523, 9.101808003889764179924612124819

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