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Properties

Label 2-2880-5.4-c1-0-14
Degree $2$
Conductor $2880$
Sign $-i$
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  + 2.23i·5-s − 4.47i·17-s + 4·19-s + 8.94i·23-s − 5.00·25-s + 8·31-s + 8.94i·47-s + 7·49-s + 4.47i·53-s − 2·61-s − 16·79-s + 17.8i·83-s + 10.0·85-s + 8.94i·95-s + 17.8i·107-s + ⋯
L(s)  = 1  + 0.999i·5-s − 1.08i·17-s + 0.917·19-s + 1.86i·23-s − 1.00·25-s + 1.43·31-s + 1.30i·47-s + 49-s + 0.614i·53-s − 0.256·61-s − 1.80·79-s + 1.96i·83-s + 1.08·85-s + 0.917i·95-s + 1.72i·107-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1729, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.625232129\)
\(L(\frac12)\) \(\approx\) \(1.625232129\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
good7 \( 1 - 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 13T^{2} \)
17 \( 1 + 4.47iT - 17T^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 - 8.94iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 8T + 31T^{2} \)
37 \( 1 - 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 - 8.94iT - 47T^{2} \)
53 \( 1 - 4.47iT - 53T^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 + 16T + 79T^{2} \)
83 \( 1 - 17.8iT - 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.161495459596230117428081717774, −7.964906876783130343120484325947, −7.44566168060990407968019347007, −6.80844689143973671556162601848, −5.90932819480809455131316243922, −5.21279738630454256812224748436, −4.14662324814721181154846216589, −3.18925479522486177479992336007, −2.57925045735441881833153553059, −1.20483941494747367304244932771, 0.57189810785759493108132960722, 1.69216114736280259383652051266, 2.82739722292712904699306670073, 3.99991164629862530909912436723, 4.62399291633909764023197961852, 5.48167141220600879897783498665, 6.22480653947314349372187394677, 7.06887888444633727762002858039, 8.108024339210010817968083700474, 8.485781057199346246616873591513

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