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Properties

Label 2-2880-60.59-c1-0-5
Degree $2$
Conductor $2880$
Sign $0.479 - 0.877i$
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  + (−0.256 − 2.22i)5-s − 3.50·7-s − 1.92·11-s − 5.50i·13-s − 4.44·17-s + 7.00i·19-s − 1.10i·23-s + (−4.86 + 1.14i)25-s + 5.47i·29-s + 8.28i·31-s + (0.900 + 7.78i)35-s − 0.778i·37-s − 2.44i·41-s + 9.55·43-s − 11.7i·47-s + ⋯
L(s)  = 1  + (−0.114 − 0.993i)5-s − 1.32·7-s − 0.581·11-s − 1.52i·13-s − 1.07·17-s + 1.60i·19-s − 0.229i·23-s + (−0.973 + 0.228i)25-s + 1.01i·29-s + 1.48i·31-s + (0.152 + 1.31i)35-s − 0.127i·37-s − 0.381i·41-s + 1.45·43-s − 1.70i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.479 - 0.877i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.479 - 0.877i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2879, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.479 - 0.877i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6527913615\)
\(L(\frac12)\) \(\approx\) \(0.6527913615\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.256 + 2.22i)T \)
good7 \( 1 + 3.50T + 7T^{2} \)
11 \( 1 + 1.92T + 11T^{2} \)
13 \( 1 + 5.50iT - 13T^{2} \)
17 \( 1 + 4.44T + 17T^{2} \)
19 \( 1 - 7.00iT - 19T^{2} \)
23 \( 1 + 1.10iT - 23T^{2} \)
29 \( 1 - 5.47iT - 29T^{2} \)
31 \( 1 - 8.28iT - 31T^{2} \)
37 \( 1 + 0.778iT - 37T^{2} \)
41 \( 1 + 2.44iT - 41T^{2} \)
43 \( 1 - 9.55T + 43T^{2} \)
47 \( 1 + 11.7iT - 47T^{2} \)
53 \( 1 - 11.5T + 53T^{2} \)
59 \( 1 - 9.78T + 59T^{2} \)
61 \( 1 - 3.45T + 61T^{2} \)
67 \( 1 + 5.45T + 67T^{2} \)
71 \( 1 - 4.25T + 71T^{2} \)
73 \( 1 - 7.27iT - 73T^{2} \)
79 \( 1 - 2.82iT - 79T^{2} \)
83 \( 1 - 4.25iT - 83T^{2} \)
89 \( 1 - 0.386iT - 89T^{2} \)
97 \( 1 - 9.29iT - 97T^{2} \)
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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

−8.715017357330477668898961223041, −8.377486920376368427087616190457, −7.40984857259048664436155221961, −6.63778854436600636937476360158, −5.59132295575148304006182035264, −5.31567321960956605761237175207, −4.03841069730897865749321387532, −3.37187288885406557108245152275, −2.33159283872033197777260204765, −0.882441988787963850168388773633, 0.25796482104273300293722961271, 2.33704877595951251345393244888, 2.71348408588424379223304981334, 3.92043235381678170361459284659, 4.49440189663524855573428543325, 5.87233542274977368027065768504, 6.47427361500836218634128674227, 7.00777204258994000089457536039, 7.66294658121567167297922651893, 8.856126330074726555817268457308

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