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Properties

Label 2-2880-16.13-c1-0-9
Degree $2$
Conductor $2880$
Sign $0.567 - 0.823i$
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 − 2.66i·7-s + (−3.49 + 3.49i)11-s + (2.94 + 2.94i)13-s − 1.85·17-s + (3.44 + 3.44i)19-s − 0.707i·23-s − 1.00i·25-s + (3.49 + 3.49i)29-s − 6.84·31-s + (−1.88 − 1.88i)35-s + (−0.0975 + 0.0975i)37-s + 10.2i·41-s + (−4.43 + 4.43i)43-s − 1.89·47-s + ⋯
L(s)  = 1  + (0.316 − 0.316i)5-s − 1.00i·7-s + (−1.05 + 1.05i)11-s + (0.815 + 0.815i)13-s − 0.448·17-s + (0.791 + 0.791i)19-s − 0.147i·23-s − 0.200i·25-s + (0.649 + 0.649i)29-s − 1.22·31-s + (−0.318 − 0.318i)35-s + (−0.0160 + 0.0160i)37-s + 1.59i·41-s + (−0.676 + 0.676i)43-s − 0.276·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.567 - 0.823i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.567 - 0.823i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (721, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.567 - 0.823i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.567217576\)
\(L(\frac12)\) \(\approx\) \(1.567217576\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.66iT - 7T^{2} \)
11 \( 1 + (3.49 - 3.49i)T - 11iT^{2} \)
13 \( 1 + (-2.94 - 2.94i)T + 13iT^{2} \)
17 \( 1 + 1.85T + 17T^{2} \)
19 \( 1 + (-3.44 - 3.44i)T + 19iT^{2} \)
23 \( 1 + 0.707iT - 23T^{2} \)
29 \( 1 + (-3.49 - 3.49i)T + 29iT^{2} \)
31 \( 1 + 6.84T + 31T^{2} \)
37 \( 1 + (0.0975 - 0.0975i)T - 37iT^{2} \)
41 \( 1 - 10.2iT - 41T^{2} \)
43 \( 1 + (4.43 - 4.43i)T - 43iT^{2} \)
47 \( 1 + 1.89T + 47T^{2} \)
53 \( 1 + (-7.43 + 7.43i)T - 53iT^{2} \)
59 \( 1 + (-0.959 + 0.959i)T - 59iT^{2} \)
61 \( 1 + (-6.49 - 6.49i)T + 61iT^{2} \)
67 \( 1 + (3.49 + 3.49i)T + 67iT^{2} \)
71 \( 1 - 7.86iT - 71T^{2} \)
73 \( 1 - 15.6iT - 73T^{2} \)
79 \( 1 - 6.70T + 79T^{2} \)
83 \( 1 + (3.87 + 3.87i)T + 83iT^{2} \)
89 \( 1 + 10.5iT - 89T^{2} \)
97 \( 1 - 4.79T + 97T^{2} \)
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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

−8.828966645490379683300975655239, −8.141087959254079472144870275381, −7.28835004432065617649963998972, −6.79461876436631377628187774324, −5.78532532457758476326514103332, −4.91871266243400673914976853408, −4.25613632247079671914850888954, −3.34419965671276174639832714973, −2.09383829628936834393632231352, −1.16907904457415940768201054528, 0.53477094156111581451136047807, 2.12805195010818537808234060510, 2.91239964141363927262962470481, 3.64102876981576148988092588169, 5.09155049365818994830687436879, 5.58865072269093378546015636483, 6.16611535064351811180295609279, 7.18709710160761429509654045557, 7.997342458769805823071415146091, 8.739943276680550372431998836085

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