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Properties

Label 2-2880-16.5-c1-0-5
Degree $2$
Conductor $2880$
Sign $-0.949 - 0.312i$
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 + 2.05i·7-s + (2.89 + 2.89i)11-s + (−0.887 + 0.887i)13-s − 7.70·17-s + (−1.96 + 1.96i)19-s + 1.75i·23-s + 1.00i·25-s + (−1.03 + 1.03i)29-s − 1.03·31-s + (−1.45 + 1.45i)35-s + (−7.76 − 7.76i)37-s − 1.08i·41-s + (4.29 + 4.29i)43-s − 8.19·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 0.776i·7-s + (0.873 + 0.873i)11-s + (−0.246 + 0.246i)13-s − 1.86·17-s + (−0.450 + 0.450i)19-s + 0.365i·23-s + 0.200i·25-s + (−0.192 + 0.192i)29-s − 0.186·31-s + (−0.245 + 0.245i)35-s + (−1.27 − 1.27i)37-s − 0.170i·41-s + (0.654 + 0.654i)43-s − 1.19·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.949 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.949 - 0.312i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.949 - 0.312i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9196304425\)
\(L(\frac12)\) \(\approx\) \(0.9196304425\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.05iT - 7T^{2} \)
11 \( 1 + (-2.89 - 2.89i)T + 11iT^{2} \)
13 \( 1 + (0.887 - 0.887i)T - 13iT^{2} \)
17 \( 1 + 7.70T + 17T^{2} \)
19 \( 1 + (1.96 - 1.96i)T - 19iT^{2} \)
23 \( 1 - 1.75iT - 23T^{2} \)
29 \( 1 + (1.03 - 1.03i)T - 29iT^{2} \)
31 \( 1 + 1.03T + 31T^{2} \)
37 \( 1 + (7.76 + 7.76i)T + 37iT^{2} \)
41 \( 1 + 1.08iT - 41T^{2} \)
43 \( 1 + (-4.29 - 4.29i)T + 43iT^{2} \)
47 \( 1 + 8.19T + 47T^{2} \)
53 \( 1 + (3.76 + 3.76i)T + 53iT^{2} \)
59 \( 1 + (3.92 + 3.92i)T + 59iT^{2} \)
61 \( 1 + (6.18 - 6.18i)T - 61iT^{2} \)
67 \( 1 + (-8.26 + 8.26i)T - 67iT^{2} \)
71 \( 1 - 6.34iT - 71T^{2} \)
73 \( 1 + 14.3iT - 73T^{2} \)
79 \( 1 - 16.1T + 79T^{2} \)
83 \( 1 + (2.72 - 2.72i)T - 83iT^{2} \)
89 \( 1 - 7.96iT - 89T^{2} \)
97 \( 1 - 7.11T + 97T^{2} \)
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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

−9.252149895612975158429719571192, −8.523075109328159265925280565593, −7.50197099302202986398773907400, −6.70293700134211022237501934257, −6.26030455274731998831485059167, −5.23332505419347797657997291658, −4.44095682913865107177200526306, −3.57356545283429504740452287636, −2.28482001895274007224510766302, −1.80722979793910012224397617804, 0.27270603678210658736956097750, 1.50127420831577385589453795528, 2.63192165409511128831636504194, 3.75897932575321857019298399791, 4.45425144757003465014848631947, 5.25869330998634842017257642520, 6.46733917006036009462913720467, 6.63058687781405661712883956081, 7.70416554967283835823879964199, 8.663695378345741214040895951302

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