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Properties

Label 2-2880-12.11-c1-0-7
Degree $2$
Conductor $2880$
Sign $-0.577 - 0.816i$
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  + i·5-s + 3.41i·7-s − 2.58·11-s + 3.41·13-s + 1.17i·17-s + 4.82·23-s − 25-s − 6i·29-s + 6.48i·31-s − 3.41·35-s + 9.07·37-s + 11.0i·41-s + 6.82i·43-s − 5.65·47-s − 4.65·49-s + ⋯
L(s)  = 1  + 0.447i·5-s + 1.29i·7-s − 0.779·11-s + 0.946·13-s + 0.284i·17-s + 1.00·23-s − 0.200·25-s − 1.11i·29-s + 1.16i·31-s − 0.577·35-s + 1.49·37-s + 1.72i·41-s + 1.04i·43-s − 0.825·47-s − 0.665·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.577 - 0.816i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1151, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.421812267\)
\(L(\frac12)\) \(\approx\) \(1.421812267\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good7 \( 1 - 3.41iT - 7T^{2} \)
11 \( 1 + 2.58T + 11T^{2} \)
13 \( 1 - 3.41T + 13T^{2} \)
17 \( 1 - 1.17iT - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 4.82T + 23T^{2} \)
29 \( 1 + 6iT - 29T^{2} \)
31 \( 1 - 6.48iT - 31T^{2} \)
37 \( 1 - 9.07T + 37T^{2} \)
41 \( 1 - 11.0iT - 41T^{2} \)
43 \( 1 - 6.82iT - 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 1.17iT - 53T^{2} \)
59 \( 1 + 6.58T + 59T^{2} \)
61 \( 1 + 12.8T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 - 5.65T + 71T^{2} \)
73 \( 1 + 10.4T + 73T^{2} \)
79 \( 1 - 14.4iT - 79T^{2} \)
83 \( 1 + 9.17T + 83T^{2} \)
89 \( 1 + 4.24iT - 89T^{2} \)
97 \( 1 - 2.48T + 97T^{2} \)
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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

−9.038231696099289746405772493310, −8.207879465118072385025112270391, −7.75379456240707310630286415821, −6.50590494439934794457572812425, −6.10401360931499072274088173993, −5.25027565534163464187060845533, −4.41997540133617260165278308774, −3.12718845327121238693493931142, −2.66618885325926136559641880625, −1.42265690545837236963947690550, 0.47199555733085493260519697418, 1.48943266048367514081448233536, 2.86524734342653966754508063740, 3.81045546073646138600453326164, 4.52942018568046865800343420785, 5.37706479002708727393670340606, 6.21214556479780484038058398828, 7.22002554933423037663167120840, 7.61112586020176316839208869697, 8.541959836183591274587314360447

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