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Properties

Label 2-2880-48.35-c1-0-23
Degree $2$
Conductor $2880$
Sign $-0.488 + 0.872i$
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 − 4.49·7-s + (4.37 − 4.37i)11-s + (1.44 + 1.44i)13-s + 6.69i·17-s + (1.91 − 1.91i)19-s + 1.18i·23-s + 1.00i·25-s + (2.16 − 2.16i)29-s − 4.28i·31-s + (3.17 + 3.17i)35-s + (0.00767 − 0.00767i)37-s + 4.93·41-s + (−2.61 − 2.61i)43-s − 12.9·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 1.69·7-s + (1.31 − 1.31i)11-s + (0.400 + 0.400i)13-s + 1.62i·17-s + (0.438 − 0.438i)19-s + 0.247i·23-s + 0.200i·25-s + (0.401 − 0.401i)29-s − 0.769i·31-s + (0.536 + 0.536i)35-s + (0.00126 − 0.00126i)37-s + 0.771·41-s + (−0.398 − 0.398i)43-s − 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.488 + 0.872i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.488 + 0.872i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (431, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.488 + 0.872i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8928465358\)
\(L(\frac12)\) \(\approx\) \(0.8928465358\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.49T + 7T^{2} \)
11 \( 1 + (-4.37 + 4.37i)T - 11iT^{2} \)
13 \( 1 + (-1.44 - 1.44i)T + 13iT^{2} \)
17 \( 1 - 6.69iT - 17T^{2} \)
19 \( 1 + (-1.91 + 1.91i)T - 19iT^{2} \)
23 \( 1 - 1.18iT - 23T^{2} \)
29 \( 1 + (-2.16 + 2.16i)T - 29iT^{2} \)
31 \( 1 + 4.28iT - 31T^{2} \)
37 \( 1 + (-0.00767 + 0.00767i)T - 37iT^{2} \)
41 \( 1 - 4.93T + 41T^{2} \)
43 \( 1 + (2.61 + 2.61i)T + 43iT^{2} \)
47 \( 1 + 12.9T + 47T^{2} \)
53 \( 1 + (6.04 + 6.04i)T + 53iT^{2} \)
59 \( 1 + (6.51 - 6.51i)T - 59iT^{2} \)
61 \( 1 + (-1.14 - 1.14i)T + 61iT^{2} \)
67 \( 1 + (-5.87 + 5.87i)T - 67iT^{2} \)
71 \( 1 + 8.93iT - 71T^{2} \)
73 \( 1 + 13.7iT - 73T^{2} \)
79 \( 1 + 4.10iT - 79T^{2} \)
83 \( 1 + (4.52 + 4.52i)T + 83iT^{2} \)
89 \( 1 + 12.3T + 89T^{2} \)
97 \( 1 + 8.38T + 97T^{2} \)
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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

−8.632690386956998795274027161960, −7.902424592811500144788717025433, −6.70042376312390487674816511321, −6.30940576711398027608378612429, −5.75683878469494946210209559476, −4.34669975042839808175712468608, −3.60737421908368597036514759966, −3.14343062789534083796624632329, −1.54875652357549943704294429277, −0.31936384729974234578071203794, 1.18613800350791781041869883426, 2.70028841936682735799844142316, 3.35293680584335220240067784191, 4.17804960488575009420189040332, 5.12980165204211586011817661746, 6.23638626474042682225725254023, 6.83873885973868840127844845415, 7.19585987860345952961216438306, 8.286467423322467238967124921531, 9.330624659916734891823112587440

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