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Properties

Label 2-2880-120.77-c1-0-16
Degree $2$
Conductor $2880$
Sign $0.326 - 0.945i$
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  + (2.11 − 0.726i)5-s + (3.61 + 3.61i)7-s − 3.26·11-s + (−3.37 − 3.37i)13-s + (−4.58 + 4.58i)17-s + 0.471·19-s + (3.32 + 3.32i)23-s + (3.94 − 3.07i)25-s + 5.98i·29-s + 5.98·31-s + (10.2 + 5.01i)35-s + (−0.639 + 0.639i)37-s + 3.39i·41-s + (3.08 + 3.08i)43-s + (−3.90 + 3.90i)47-s + ⋯
L(s)  = 1  + (0.945 − 0.324i)5-s + (1.36 + 1.36i)7-s − 0.984·11-s + (−0.937 − 0.937i)13-s + (−1.11 + 1.11i)17-s + 0.108·19-s + (0.693 + 0.693i)23-s + (0.788 − 0.614i)25-s + 1.11i·29-s + 1.07·31-s + (1.73 + 0.847i)35-s + (−0.105 + 0.105i)37-s + 0.530i·41-s + (0.470 + 0.470i)43-s + (−0.569 + 0.569i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.326 - 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.326 - 0.945i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (737, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.326 - 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.131713370\)
\(L(\frac12)\) \(\approx\) \(2.131713370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.11 + 0.726i)T \)
good7 \( 1 + (-3.61 - 3.61i)T + 7iT^{2} \)
11 \( 1 + 3.26T + 11T^{2} \)
13 \( 1 + (3.37 + 3.37i)T + 13iT^{2} \)
17 \( 1 + (4.58 - 4.58i)T - 17iT^{2} \)
19 \( 1 - 0.471T + 19T^{2} \)
23 \( 1 + (-3.32 - 3.32i)T + 23iT^{2} \)
29 \( 1 - 5.98iT - 29T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 + (0.639 - 0.639i)T - 37iT^{2} \)
41 \( 1 - 3.39iT - 41T^{2} \)
43 \( 1 + (-3.08 - 3.08i)T + 43iT^{2} \)
47 \( 1 + (3.90 - 3.90i)T - 47iT^{2} \)
53 \( 1 + (-1.26 + 1.26i)T - 53iT^{2} \)
59 \( 1 - 12.2iT - 59T^{2} \)
61 \( 1 - 10.6iT - 61T^{2} \)
67 \( 1 + (-10.3 + 10.3i)T - 67iT^{2} \)
71 \( 1 + 9.55iT - 71T^{2} \)
73 \( 1 + (-3.92 + 3.92i)T - 73iT^{2} \)
79 \( 1 + 1.87iT - 79T^{2} \)
83 \( 1 + (-5.24 + 5.24i)T - 83iT^{2} \)
89 \( 1 - 12.0T + 89T^{2} \)
97 \( 1 + (-5.34 - 5.34i)T + 97iT^{2} \)
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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

−8.893404714016338215991661582124, −8.202823138774539588355553827703, −7.68686677257590649213928428994, −6.46059694404378867777776926060, −5.65241722226703193729005604855, −5.10218511427431651791755639515, −4.64580785770309790794934581736, −2.89284523528127858240727983483, −2.30618024643272762988139178120, −1.39581117295178418849898607633, 0.66149664455381271219314363214, 2.04169501463680498831564122561, 2.57125357996087762026441791825, 4.07671095988765120089375263175, 4.87506993421944187408361221622, 5.20852679980798403718162462674, 6.68065507692579205180158422779, 6.98190807959519021353120853468, 7.81245547268814744102194135818, 8.540696574794544307239111581233

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