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Properties

Label 2-2880-15.8-c1-0-40
Degree $2$
Conductor $2880$
Sign $-0.999 + 0.000538i$
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  + (0.137 − 2.23i)5-s + (−0.806 + 0.806i)7-s + 5.87i·11-s + (0.193 + 0.193i)13-s + (−3.50 − 3.50i)17-s + 2.38i·19-s + (4.18 − 4.18i)23-s + (−4.96 − 0.612i)25-s − 7.29·29-s + 4.31·31-s + (1.68 + 1.90i)35-s + (−1.54 + 1.54i)37-s − 3.32i·41-s + (−8.31 − 8.31i)43-s + (−6.42 − 6.42i)47-s + ⋯
L(s)  = 1  + (0.0613 − 0.998i)5-s + (−0.304 + 0.304i)7-s + 1.77i·11-s + (0.0537 + 0.0537i)13-s + (−0.851 − 0.851i)17-s + 0.547i·19-s + (0.873 − 0.873i)23-s + (−0.992 − 0.122i)25-s − 1.35·29-s + 0.774·31-s + (0.285 + 0.322i)35-s + (−0.253 + 0.253i)37-s − 0.519i·41-s + (−1.26 − 1.26i)43-s + (−0.937 − 0.937i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.000538i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.000538i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.999 + 0.000538i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2753, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ -0.999 + 0.000538i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1529598686\)
\(L(\frac12)\) \(\approx\) \(0.1529598686\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.137 + 2.23i)T \)
good7 \( 1 + (0.806 - 0.806i)T - 7iT^{2} \)
11 \( 1 - 5.87iT - 11T^{2} \)
13 \( 1 + (-0.193 - 0.193i)T + 13iT^{2} \)
17 \( 1 + (3.50 + 3.50i)T + 17iT^{2} \)
19 \( 1 - 2.38iT - 19T^{2} \)
23 \( 1 + (-4.18 + 4.18i)T - 23iT^{2} \)
29 \( 1 + 7.29T + 29T^{2} \)
31 \( 1 - 4.31T + 31T^{2} \)
37 \( 1 + (1.54 - 1.54i)T - 37iT^{2} \)
41 \( 1 + 3.32iT - 41T^{2} \)
43 \( 1 + (8.31 + 8.31i)T + 43iT^{2} \)
47 \( 1 + (6.42 + 6.42i)T + 47iT^{2} \)
53 \( 1 + (-4.64 + 4.64i)T - 53iT^{2} \)
59 \( 1 + 10.4T + 59T^{2} \)
61 \( 1 + 7.53T + 61T^{2} \)
67 \( 1 + (4.31 - 4.31i)T - 67iT^{2} \)
71 \( 1 + 9.47iT - 71T^{2} \)
73 \( 1 + (-0.0376 - 0.0376i)T + 73iT^{2} \)
79 \( 1 + 3.68iT - 79T^{2} \)
83 \( 1 + (0.221 - 0.221i)T - 83iT^{2} \)
89 \( 1 + 9.52T + 89T^{2} \)
97 \( 1 + (8.27 - 8.27i)T - 97iT^{2} \)
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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

−8.593815261619859635919316574515, −7.57077127146003847183395749131, −6.96631392501829168204348245100, −6.11542870050667667143707178671, −4.98237317939572234638871312536, −4.74318049732552533057187312535, −3.70412471896301751826617835123, −2.40367894770227961598272253597, −1.61489301129091999097055696163, −0.04633895723086844495443861016, 1.50903898606169923662101155155, 2.92630677552217553256477262799, 3.33175596489308093978780203299, 4.31046046066274687912329230017, 5.51710073314101546077108120888, 6.21621307937725215151927447380, 6.74874157962649068084254126660, 7.63305246992867453409151791661, 8.372291967037002902386883066085, 9.138800045625011350910226699001

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