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Properties

Label 2-2880-16.5-c1-0-22
Degree $2$
Conductor $2880$
Sign $0.541 - 0.841i$
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 + 4.35i·7-s + (2.37 + 2.37i)11-s + (3.66 − 3.66i)13-s + 5.50·17-s + (−0.0623 + 0.0623i)19-s − 2.71i·23-s + 1.00i·25-s + (2.48 − 2.48i)29-s + 6.29·31-s + (−3.07 + 3.07i)35-s + (−6.02 − 6.02i)37-s + 1.43i·41-s + (−0.185 − 0.185i)43-s + 4.11·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.64i·7-s + (0.715 + 0.715i)11-s + (1.01 − 1.01i)13-s + 1.33·17-s + (−0.0142 + 0.0142i)19-s − 0.565i·23-s + 0.200i·25-s + (0.462 − 0.462i)29-s + 1.13·31-s + (−0.520 + 0.520i)35-s + (−0.990 − 0.990i)37-s + 0.224i·41-s + (−0.0283 − 0.0283i)43-s + 0.599·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.541 - 0.841i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.541 - 0.841i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.541 - 0.841i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.541 - 0.841i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.325155695\)
\(L(\frac12)\) \(\approx\) \(2.325155695\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.35iT - 7T^{2} \)
11 \( 1 + (-2.37 - 2.37i)T + 11iT^{2} \)
13 \( 1 + (-3.66 + 3.66i)T - 13iT^{2} \)
17 \( 1 - 5.50T + 17T^{2} \)
19 \( 1 + (0.0623 - 0.0623i)T - 19iT^{2} \)
23 \( 1 + 2.71iT - 23T^{2} \)
29 \( 1 + (-2.48 + 2.48i)T - 29iT^{2} \)
31 \( 1 - 6.29T + 31T^{2} \)
37 \( 1 + (6.02 + 6.02i)T + 37iT^{2} \)
41 \( 1 - 1.43iT - 41T^{2} \)
43 \( 1 + (0.185 + 0.185i)T + 43iT^{2} \)
47 \( 1 - 4.11T + 47T^{2} \)
53 \( 1 + (-9.16 - 9.16i)T + 53iT^{2} \)
59 \( 1 + (5.17 + 5.17i)T + 59iT^{2} \)
61 \( 1 + (-7.00 + 7.00i)T - 61iT^{2} \)
67 \( 1 + (3.40 - 3.40i)T - 67iT^{2} \)
71 \( 1 - 12.9iT - 71T^{2} \)
73 \( 1 - 6.32iT - 73T^{2} \)
79 \( 1 + 10.6T + 79T^{2} \)
83 \( 1 + (3.00 - 3.00i)T - 83iT^{2} \)
89 \( 1 - 4.12iT - 89T^{2} \)
97 \( 1 + 8.15T + 97T^{2} \)
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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

−8.720572180293947456510281580088, −8.389474937222660107920276543577, −7.39364223378167236975242713448, −6.44060463781897920281410638995, −5.78930380136885109013275751561, −5.31057418179403206337870785804, −4.10796255017951073055519462128, −3.08235580039436997773696576281, −2.36931393355087780682729445588, −1.19973469785520737248300301094, 0.935560070765915646022947042597, 1.50297338295236521595828919890, 3.25013402811573497520762991733, 3.84710275197642136160203048257, 4.60589384722477066285506128423, 5.64937414417567727023464605155, 6.48699807441991083997361495261, 7.03580777877297856100306049667, 7.936068295665508922032122126282, 8.658423003896742636512576230800

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