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Properties

Label 2-2880-5.2-c0-0-2
Degree $2$
Conductor $2880$
Sign $-0.850 + 0.525i$
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  − 5-s + (−1 + i)13-s + (−1 − i)17-s + 25-s − 2i·29-s + (−1 − i)37-s − 2·41-s − i·49-s + (−1 + i)53-s + (1 − i)65-s + (−1 + i)73-s + (1 + i)85-s + 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯
L(s)  = 1  − 5-s + (−1 + i)13-s + (−1 − i)17-s + 25-s − 2i·29-s + (−1 − i)37-s − 2·41-s − i·49-s + (−1 + i)53-s + (1 − i)65-s + (−1 + i)73-s + (1 + i)85-s + 2i·89-s + (−1 − i)97-s − 2i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.850 + 0.525i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (577, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ -0.850 + 0.525i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.2437904118\)
\(L(\frac12)\) \(\approx\) \(0.2437904118\)
\(L(1)\) 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 \( 1 + T \)
good7 \( 1 + iT^{2} \)
11 \( 1 + T^{2} \)
13 \( 1 + (1 - i)T - iT^{2} \)
17 \( 1 + (1 + i)T + iT^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 - iT^{2} \)
29 \( 1 + 2iT - T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 + (1 + i)T + iT^{2} \)
41 \( 1 + 2T + T^{2} \)
43 \( 1 - iT^{2} \)
47 \( 1 + iT^{2} \)
53 \( 1 + (1 - i)T - iT^{2} \)
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 \( 1 - 2iT - T^{2} \)
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

−8.623345293231622288704976911294, −7.87050787384386015494758656003, −7.04464824917153803179918194201, −6.70696618814236765793948525271, −5.42282872854658132784515926932, −4.57197749453620855803271023342, −4.05979962730519690993546485919, −2.92358627559292952382383428145, −1.98397832221700599696582231182, −0.14623722214497856810189978782, 1.62252415666222125032276672748, 2.98002060769753690289434480474, 3.59614965549997602557157618609, 4.71980958619243795448462942401, 5.16618443262300700895108142875, 6.41112449763050548042067858941, 7.04927263170452841074601809363, 7.80131188430928146097230568953, 8.483496274461369790474295068571, 9.028902312417474995090016858901

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