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Properties

Label 2-2880-48.35-c1-0-24
Degree $2$
Conductor $2880$
Sign $0.371 + 0.928i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.371 + 0.928i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.371 + 0.928i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.371 + 0.928i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.695872077\)
\(L(\frac12)\) \(\approx\) \(1.695872077\)
\(L(\frac{3}{2})\) 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

−8.667676784893084159823511732291, −7.65327101648905813587998240888, −7.46681062840536622954430191132, −6.39668231885625332912437020209, −5.17858024324726043086953397226, −4.94441364813276455817907416718, −3.98975896911922614562949670692, −2.84186135670247498568660976661, −1.87346580228100719108769026834, −0.59577207499923470055280335853, 1.18781610186388405136184142887, 2.22476200438468500420868262879, 3.43577129223645632255349248279, 4.03381145695721196497832602624, 5.28126784047358831468482429297, 5.60176487426200317727453618712, 6.74715485435560893698041685269, 7.56327375094765049570673210692, 8.126865008760519591567949116879, 8.688962262634904779111070477183

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