Properties

Label 2-2880-15.14-c0-0-3
Degree $2$
Conductor $2880$
Sign $0.169 + 0.985i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s − 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s − 1.41·53-s + 2·61-s + (−1.41 − 1.41i)65-s + 2i·73-s + (−1.00 + 1.00i)85-s − 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s − 2i·13-s − 1.41·17-s − 1.00i·25-s − 1.41i·29-s + 1.41i·41-s + 49-s − 1.41·53-s + 2·61-s + (−1.41 − 1.41i)65-s + 2i·73-s + (−1.00 + 1.00i)85-s − 1.41i·89-s − 2i·97-s + 1.41i·101-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.169 + 0.985i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (449, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.169 + 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.250272320\)
\(L(\frac12)\) \(\approx\) \(1.250272320\)
\(L(1)\) 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 \( 1 + (-0.707 + 0.707i)T \)
good7 \( 1 - T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 2iT - T^{2} \)
17 \( 1 + 1.41T + T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + 1.41iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - T^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + 1.41T + T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 - 2T + T^{2} \)
67 \( 1 - T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 - 2iT - T^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + 1.41iT - T^{2} \)
97 \( 1 + 2iT - T^{2} \)
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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

−8.597406698164618218530687850489, −8.272457406295000043022029540664, −7.35194979139766840929829953786, −6.30687507491367460525312548754, −5.74619372932479945664830337675, −4.96246540582496780540651216639, −4.19069659732807037582480196262, −2.96741114604231083956354136386, −2.13073161634101885743960030280, −0.77243171940817987685437170764, 1.74922808533727725495448654560, 2.35494142730473124917319451416, 3.55450206630268070716617230156, 4.41745089041717738587916943047, 5.27831601646393699416224068562, 6.31881907363701022271659036266, 6.79357097658849185537644977339, 7.32779912983168513973858306075, 8.646350057116396230430802559296, 9.113097000047503054415477516749

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