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Properties

Label 2-2880-16.5-c1-0-30
Degree $2$
Conductor $2880$
Sign $0.0450 + 0.998i$
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 − 2.18i·7-s + (−0.00889 − 0.00889i)11-s + (1.72 − 1.72i)13-s + 5.54·17-s + (4.94 − 4.94i)19-s + 3.01i·23-s + 1.00i·25-s + (−3.20 + 3.20i)29-s − 3.58·31-s + (−1.54 + 1.54i)35-s + (4.97 + 4.97i)37-s + 3.76i·41-s + (−6.81 − 6.81i)43-s − 10.0·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 0.824i·7-s + (−0.00268 − 0.00268i)11-s + (0.479 − 0.479i)13-s + 1.34·17-s + (1.13 − 1.13i)19-s + 0.628i·23-s + 0.200i·25-s + (−0.595 + 0.595i)29-s − 0.643·31-s + (−0.260 + 0.260i)35-s + (0.818 + 0.818i)37-s + 0.587i·41-s + (−1.03 − 1.03i)43-s − 1.47·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0450 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.0450 + 0.998i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.0450 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.663016748\)
\(L(\frac12)\) \(\approx\) \(1.663016748\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.18iT - 7T^{2} \)
11 \( 1 + (0.00889 + 0.00889i)T + 11iT^{2} \)
13 \( 1 + (-1.72 + 1.72i)T - 13iT^{2} \)
17 \( 1 - 5.54T + 17T^{2} \)
19 \( 1 + (-4.94 + 4.94i)T - 19iT^{2} \)
23 \( 1 - 3.01iT - 23T^{2} \)
29 \( 1 + (3.20 - 3.20i)T - 29iT^{2} \)
31 \( 1 + 3.58T + 31T^{2} \)
37 \( 1 + (-4.97 - 4.97i)T + 37iT^{2} \)
41 \( 1 - 3.76iT - 41T^{2} \)
43 \( 1 + (6.81 + 6.81i)T + 43iT^{2} \)
47 \( 1 + 10.0T + 47T^{2} \)
53 \( 1 + (-0.932 - 0.932i)T + 53iT^{2} \)
59 \( 1 + (4.60 + 4.60i)T + 59iT^{2} \)
61 \( 1 + (-4.17 + 4.17i)T - 61iT^{2} \)
67 \( 1 + (-11.0 + 11.0i)T - 67iT^{2} \)
71 \( 1 + 12.1iT - 71T^{2} \)
73 \( 1 + 7.12iT - 73T^{2} \)
79 \( 1 - 3.41T + 79T^{2} \)
83 \( 1 + (-5.31 + 5.31i)T - 83iT^{2} \)
89 \( 1 - 5.06iT - 89T^{2} \)
97 \( 1 + 10.3T + 97T^{2} \)
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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

−8.476190431402728022700095348422, −7.74452063048885347402916974490, −7.28133923687082096356256198477, −6.35737740293776924997172798231, −5.32068767041646087216660964992, −4.83391819272241219893674597553, −3.55651190497759234692973995291, −3.24689818744372411917655665437, −1.57146851821581514189316336622, −0.60478168559560135005071257631, 1.22931141462169397800295418227, 2.39795883540876112670448524628, 3.39002674115245429725470018401, 4.05676140501820723418824836475, 5.33259601940749111605005558225, 5.77596164021887952936890810222, 6.66397920050822971938847237330, 7.59022591678251502728013930148, 8.119844579609787449998170869182, 8.923266021726920026055618112031

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