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Properties

Label 2-2880-48.35-c1-0-0
Degree $2$
Conductor $2880$
Sign $-0.857 - 0.513i$
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 + 1.80·7-s + (−0.135 + 0.135i)11-s + (−4.41 − 4.41i)13-s − 3.38i·17-s + (−3.50 + 3.50i)19-s − 5.73i·23-s + 1.00i·25-s + (−6.69 + 6.69i)29-s + 10.6i·31-s + (−1.27 − 1.27i)35-s + (−2.24 + 2.24i)37-s + 2.15·41-s + (8.06 + 8.06i)43-s − 0.779·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s + 0.680·7-s + (−0.0409 + 0.0409i)11-s + (−1.22 − 1.22i)13-s − 0.820i·17-s + (−0.804 + 0.804i)19-s − 1.19i·23-s + 0.200i·25-s + (−1.24 + 1.24i)29-s + 1.90i·31-s + (−0.215 − 0.215i)35-s + (−0.368 + 0.368i)37-s + 0.336·41-s + (1.22 + 1.22i)43-s − 0.113·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 - 0.513i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.857 - 0.513i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.857 - 0.513i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1409716215\)
\(L(\frac12)\) \(\approx\) \(0.1409716215\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.80T + 7T^{2} \)
11 \( 1 + (0.135 - 0.135i)T - 11iT^{2} \)
13 \( 1 + (4.41 + 4.41i)T + 13iT^{2} \)
17 \( 1 + 3.38iT - 17T^{2} \)
19 \( 1 + (3.50 - 3.50i)T - 19iT^{2} \)
23 \( 1 + 5.73iT - 23T^{2} \)
29 \( 1 + (6.69 - 6.69i)T - 29iT^{2} \)
31 \( 1 - 10.6iT - 31T^{2} \)
37 \( 1 + (2.24 - 2.24i)T - 37iT^{2} \)
41 \( 1 - 2.15T + 41T^{2} \)
43 \( 1 + (-8.06 - 8.06i)T + 43iT^{2} \)
47 \( 1 + 0.779T + 47T^{2} \)
53 \( 1 + (5.61 + 5.61i)T + 53iT^{2} \)
59 \( 1 + (4.77 - 4.77i)T - 59iT^{2} \)
61 \( 1 + (-4.75 - 4.75i)T + 61iT^{2} \)
67 \( 1 + (1.44 - 1.44i)T - 67iT^{2} \)
71 \( 1 + 9.77iT - 71T^{2} \)
73 \( 1 - 1.76iT - 73T^{2} \)
79 \( 1 + 8.26iT - 79T^{2} \)
83 \( 1 + (1.69 + 1.69i)T + 83iT^{2} \)
89 \( 1 + 7.74T + 89T^{2} \)
97 \( 1 + 8.62T + 97T^{2} \)
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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

−8.944142820811898085477434566101, −8.295329309473647444858924374629, −7.62695513888048700043086552780, −7.01291004705263585660935721941, −5.93386695985323878858462390972, −4.98169568184076453368031295743, −4.69650487789507250450613486501, −3.43449927199162334817965508368, −2.56035414509308469407908616670, −1.36432893532253759307242708749, 0.04315257436281866878823648264, 1.84281334455488229382302213694, 2.46965674031628433278244888505, 3.94346885844824469384657039492, 4.32554726640050448919640343790, 5.37287242049873104596336735012, 6.17992994492576173101921609977, 7.12691755418347657115794047931, 7.63467995654838981585212535370, 8.347565730531359441657140999715

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