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Properties

Label 2-2880-60.47-c0-0-1
Degree $2$
Conductor $2880$
Sign $0.927 - 0.374i$
Analytic cond. $1.43730$
Root an. cond. $1.19887$
Motivic weight $0$
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 + (1 − i)13-s − 1.00i·25-s + 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + 1.41i·65-s + (1 − i)73-s + 1.41·89-s + (1 + i)97-s + 1.41i·101-s + (−1.41 − 1.41i)113-s + ⋯
L(s)  = 1  + (−0.707 + 0.707i)5-s + (1 − i)13-s − 1.00i·25-s + 1.41·29-s + (1 + i)37-s + 1.41i·41-s + i·49-s + 1.41i·65-s + (1 − i)73-s + 1.41·89-s + (1 + i)97-s + 1.41i·101-s + (−1.41 − 1.41i)113-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.927 - 0.374i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.927 - 0.374i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1727, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.927 - 0.374i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.134723761\)
\(L(\frac12)\) \(\approx\) \(1.134723761\)
\(L(1)\) 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 \( 1 - iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (-1 + i)T - iT^{2} \)
17 \( 1 - iT^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 - 1.41T + T^{2} \)
31 \( 1 - T^{2} \)
37 \( 1 + (-1 - i)T + iT^{2} \)
41 \( 1 - 1.41iT - T^{2} \)
43 \( 1 + iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + iT^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - iT^{2} \)
71 \( 1 + T^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 + T^{2} \)
83 \( 1 - iT^{2} \)
89 \( 1 - 1.41T + T^{2} \)
97 \( 1 + (-1 - i)T + iT^{2} \)
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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

−8.873465601658402769851660347358, −7.994609935243394402471976592060, −7.79113241438711720063752997172, −6.52854087272630818891897422162, −6.24980740707919572842893286960, −5.06964592028473578836772658584, −4.20970502947956217005523135067, −3.29393004990028254972439072224, −2.68022158481788757748551046625, −1.06977353466311509555269216726, 0.961147639070557651401061248969, 2.17365229747560957090164627978, 3.51484269255637668306544513612, 4.15289863695008864225571618795, 4.91719727982237212901928681010, 5.85047432093224294999581749082, 6.68460574457802257040341998867, 7.44562025115429507810881567381, 8.314984094297826274808098409995, 8.796298838334247651592337748034

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