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Properties

Label 2-2880-120.77-c1-0-11
Degree $2$
Conductor $2880$
Sign $0.981 - 0.189i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.981 - 0.189i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.981 - 0.189i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.981 - 0.189i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7170406987\)
\(L(\frac12)\) \(\approx\) \(0.7170406987\)
\(L(\frac{3}{2})\) 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

−8.719115014023735345315368264439, −7.929991590450376564713765661201, −7.29486504269275437957472696989, −6.48998708546621755378278777606, −6.12340649588313588360482611998, −4.52940894263723476687204141918, −3.96930185548127024578628284031, −3.41993549886856643192743643879, −2.22046349419798279067213797450, −0.55287632116308786681153818212, 0.43875800475114073259303501880, 2.29646978768521229994580221220, 3.14646501455918871301047071377, 3.75340582110234486122357935003, 5.10431612454244537398591387504, 5.55097475918717211528239782831, 6.45211702938023628982165198430, 7.30239781073847941441464441229, 8.046982794112551706419115085846, 8.939431779816697272178987876179

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