Properties

Label 2-2880-60.23-c0-0-2
Degree $2$
Conductor $2880$
Sign $0.749 + 0.662i$
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 + (1.41 + 1.41i)17-s − 1.00i·25-s + 1.41·29-s + (1 − i)37-s − 1.41i·41-s i·49-s + (−1.41 + 1.41i)53-s − 1.41·65-s + (−1 − i)73-s + 2.00·85-s + 1.41·89-s + (−1 + i)97-s + 1.41i·101-s + ⋯
L(s)  = 1  + (0.707 − 0.707i)5-s + (−1 − i)13-s + (1.41 + 1.41i)17-s − 1.00i·25-s + 1.41·29-s + (1 − i)37-s − 1.41i·41-s i·49-s + (−1.41 + 1.41i)53-s − 1.41·65-s + (−1 − i)73-s + 2.00·85-s + 1.41·89-s + (−1 + i)97-s + 1.41i·101-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 + 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.749 + 0.662i$
Analytic conductor: \(1.43730\)
Root analytic conductor: \(1.19887\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2303, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :0),\ 0.749 + 0.662i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.403671547\)
\(L(\frac12)\) \(\approx\) \(1.403671547\)
\(L(1)\) 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 \( 1 + (-1.41 - 1.41i)T + iT^{2} \)
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 \( 1 + (1.41 - 1.41i)T - iT^{2} \)
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 \( 1 - 1.41T + T^{2} \)
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

−8.890771953579775785733471181651, −8.051053988755101448938724834908, −7.60403561805083509603135668208, −6.41054305811444356417102486091, −5.71841996654793039583486992997, −5.14497657420040012058525392844, −4.22920663736593155309763152055, −3.15546935633479616241628653306, −2.15744213240418502592389045467, −1.00487331015154251525109507367, 1.38687071334865829826438095344, 2.63691476428635955911080118077, 3.11611641826183295277046956250, 4.52546195238292836567902063255, 5.11986727781763094305097409929, 6.11996903532565470714220458433, 6.76622273291187949016051496385, 7.43822296514295709863452987168, 8.186695361437362424015718446474, 9.350671075800394598446294603861

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