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Properties

Label 2-2880-48.35-c1-0-20
Degree $2$
Conductor $2880$
Sign $0.999 - 0.0404i$
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 + 2.69·7-s + (1.63 − 1.63i)11-s + (0.257 + 0.257i)13-s + 3.62i·17-s + (3.93 − 3.93i)19-s + 3.14i·23-s + 1.00i·25-s + (4.00 − 4.00i)29-s − 8.73i·31-s + (1.90 + 1.90i)35-s + (−2.71 + 2.71i)37-s + 5.22·41-s + (3.76 + 3.76i)43-s − 8.54·47-s + ⋯
L(s)  = 1  + (0.316 + 0.316i)5-s + 1.01·7-s + (0.491 − 0.491i)11-s + (0.0714 + 0.0714i)13-s + 0.879i·17-s + (0.902 − 0.902i)19-s + 0.655i·23-s + 0.200i·25-s + (0.744 − 0.744i)29-s − 1.56i·31-s + (0.321 + 0.321i)35-s + (−0.446 + 0.446i)37-s + 0.815·41-s + (0.574 + 0.574i)43-s − 1.24·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0404i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0404i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.999 - 0.0404i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.999 - 0.0404i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.423814427\)
\(L(\frac12)\) \(\approx\) \(2.423814427\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.69T + 7T^{2} \)
11 \( 1 + (-1.63 + 1.63i)T - 11iT^{2} \)
13 \( 1 + (-0.257 - 0.257i)T + 13iT^{2} \)
17 \( 1 - 3.62iT - 17T^{2} \)
19 \( 1 + (-3.93 + 3.93i)T - 19iT^{2} \)
23 \( 1 - 3.14iT - 23T^{2} \)
29 \( 1 + (-4.00 + 4.00i)T - 29iT^{2} \)
31 \( 1 + 8.73iT - 31T^{2} \)
37 \( 1 + (2.71 - 2.71i)T - 37iT^{2} \)
41 \( 1 - 5.22T + 41T^{2} \)
43 \( 1 + (-3.76 - 3.76i)T + 43iT^{2} \)
47 \( 1 + 8.54T + 47T^{2} \)
53 \( 1 + (-2.25 - 2.25i)T + 53iT^{2} \)
59 \( 1 + (3.43 - 3.43i)T - 59iT^{2} \)
61 \( 1 + (-8.79 - 8.79i)T + 61iT^{2} \)
67 \( 1 + (-5.07 + 5.07i)T - 67iT^{2} \)
71 \( 1 - 4.48iT - 71T^{2} \)
73 \( 1 + 12.9iT - 73T^{2} \)
79 \( 1 + 12.5iT - 79T^{2} \)
83 \( 1 + (1.25 + 1.25i)T + 83iT^{2} \)
89 \( 1 + 15.8T + 89T^{2} \)
97 \( 1 - 11.7T + 97T^{2} \)
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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

−8.757965780391729924714089511364, −7.979063970820300556505375716116, −7.40406012103022935619915564005, −6.37549979997057513734327414731, −5.80596451761854479536596490563, −4.87516009383611797592905688709, −4.08737013614333655492419588829, −3.07905278762704581380280178178, −2.03638676758807845943533024276, −1.01189665024968800253208685244, 1.05975717027461354362923527639, 1.90861493602405113113486897402, 3.07239117287911649731708554372, 4.14541892913050549631117772191, 5.02472188709991008220302925488, 5.43026632733877741005040259621, 6.62679063908534324933489651859, 7.21561279946150983316883942615, 8.156388546464620973243283760833, 8.655392814684356010457240264302

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