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Properties

Label 2-2880-16.5-c1-0-26
Degree $2$
Conductor $2880$
Sign $0.914 + 0.403i$
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  + (0.707 + 0.707i)5-s − 4.30i·7-s + (4.45 + 4.45i)11-s + (−0.918 + 0.918i)13-s + 2.39·17-s + (5.30 − 5.30i)19-s − 2.19i·23-s + 1.00i·25-s + (−4.30 + 4.30i)29-s − 5.39·31-s + (3.04 − 3.04i)35-s + (0.841 + 0.841i)37-s − 6.87i·41-s + (6.63 + 6.63i)43-s + 11.0·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s − 1.62i·7-s + (1.34 + 1.34i)11-s + (−0.254 + 0.254i)13-s + 0.580·17-s + (1.21 − 1.21i)19-s − 0.458i·23-s + 0.200i·25-s + (−0.798 + 0.798i)29-s − 0.969·31-s + (0.515 − 0.515i)35-s + (0.138 + 0.138i)37-s − 1.07i·41-s + (1.01 + 1.01i)43-s + 1.60·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.914 + 0.403i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.914 + 0.403i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.914 + 0.403i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.206823060\)
\(L(\frac12)\) \(\approx\) \(2.206823060\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.707 - 0.707i)T \)
good7 \( 1 + 4.30iT - 7T^{2} \)
11 \( 1 + (-4.45 - 4.45i)T + 11iT^{2} \)
13 \( 1 + (0.918 - 0.918i)T - 13iT^{2} \)
17 \( 1 - 2.39T + 17T^{2} \)
19 \( 1 + (-5.30 + 5.30i)T - 19iT^{2} \)
23 \( 1 + 2.19iT - 23T^{2} \)
29 \( 1 + (4.30 - 4.30i)T - 29iT^{2} \)
31 \( 1 + 5.39T + 31T^{2} \)
37 \( 1 + (-0.841 - 0.841i)T + 37iT^{2} \)
41 \( 1 + 6.87iT - 41T^{2} \)
43 \( 1 + (-6.63 - 6.63i)T + 43iT^{2} \)
47 \( 1 - 11.0T + 47T^{2} \)
53 \( 1 + (-0.600 - 0.600i)T + 53iT^{2} \)
59 \( 1 + (-0.850 - 0.850i)T + 59iT^{2} \)
61 \( 1 + (7.50 - 7.50i)T - 61iT^{2} \)
67 \( 1 + (-8.00 + 8.00i)T - 67iT^{2} \)
71 \( 1 + 2.29iT - 71T^{2} \)
73 \( 1 - 1.27iT - 73T^{2} \)
79 \( 1 + 8.05T + 79T^{2} \)
83 \( 1 + (0.110 - 0.110i)T - 83iT^{2} \)
89 \( 1 + 5.90iT - 89T^{2} \)
97 \( 1 - 11.0T + 97T^{2} \)
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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

−9.027853830103215551413577880577, −7.49297662493683336951926100776, −7.26896356585173742719014851480, −6.77235453676057542281781075744, −5.66285222083832345918040646121, −4.61293509516476426201157699759, −4.06984711383754749382291827490, −3.16608898339204922639517315249, −1.85068174123714929943719479530, −0.895304400821604645036416366030, 1.05998105495557943224974651477, 2.11853233482001393440768607278, 3.21608874858466010508684449037, 3.89658104015606059568812371857, 5.33768406037036953311041004205, 5.73355009852061281910410689250, 6.18538756963763581273368096833, 7.45266964210839288117687684680, 8.178366755321011249524518473599, 9.021139185554866955867445007831

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