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Properties

Label 2-2880-60.23-c0-0-0
Degree $2$
Conductor $2880$
Sign $0.662 - 0.749i$
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

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

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.662 - 0.749i$
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: \((2,\ 2880,\ (\ :0),\ 0.662 - 0.749i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.437834302\)
\(L(\frac12)\) \(\approx\) \(1.437834302\)
\(L(1)\) 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

−9.220758420086748418565222567411, −8.349363461794503287421435372998, −7.44181326082767250668674177415, −6.70664653852982943578260943930, −6.08448862648407910074860592399, −5.38154692373228313663656558960, −4.23006908493572465921853690793, −3.46909484873863335559908819142, −2.39562164731800130786786219744, −1.51076933828162805237097319205, 1.00092017584885616291317561656, 2.08076936946938008534206277094, 3.21236202764935327546139350673, 4.14549755270071257599407314644, 5.12146484956554561611566709718, 5.78296567434150685505480316884, 6.35759006020709001070846881534, 7.48082322806505787024127677068, 8.181028068706345065031619576731, 8.885990131864725787340277399671

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