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Properties

Label 2-2880-15.8-c1-0-22
Degree $2$
Conductor $2880$
Sign $0.749 - 0.662i$
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  + (2.12 − 0.707i)5-s + (5 + 5i)13-s + (5.65 + 5.65i)17-s + (3.99 − 3i)25-s − 9.89·29-s + (−7 + 7i)37-s − 1.41i·41-s + 7i·49-s + (2.82 − 2.82i)53-s − 12·61-s + (14.1 + 7.07i)65-s + (5 + 5i)73-s + (16 + 7.99i)85-s + 18.3·89-s + (5 − 5i)97-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)5-s + (1.38 + 1.38i)13-s + (1.37 + 1.37i)17-s + (0.799 − 0.600i)25-s − 1.83·29-s + (−1.15 + 1.15i)37-s − 0.220i·41-s + i·49-s + (0.388 − 0.388i)53-s − 1.53·61-s + (1.75 + 0.877i)65-s + (0.585 + 0.585i)73-s + (1.73 + 0.867i)85-s + 1.94·89-s + (0.507 − 0.507i)97-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.749 - 0.662i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.749 - 0.662i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.749 - 0.662i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.371226424\)
\(L(\frac12)\) \(\approx\) \(2.371226424\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.12 + 0.707i)T \)
good7 \( 1 - 7iT^{2} \)
11 \( 1 - 11T^{2} \)
13 \( 1 + (-5 - 5i)T + 13iT^{2} \)
17 \( 1 + (-5.65 - 5.65i)T + 17iT^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 9.89T + 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 + (7 - 7i)T - 37iT^{2} \)
41 \( 1 + 1.41iT - 41T^{2} \)
43 \( 1 + 43iT^{2} \)
47 \( 1 + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 + 59T^{2} \)
61 \( 1 + 12T + 61T^{2} \)
67 \( 1 - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 - 83iT^{2} \)
89 \( 1 - 18.3T + 89T^{2} \)
97 \( 1 + (-5 + 5i)T - 97iT^{2} \)
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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

−8.900520474090341955057200736063, −8.280664899938661702277166509479, −7.33116575672126177315034488654, −6.31168141018345505940774537600, −5.96800505537607429534696568796, −5.10089946298804020007157281246, −4.04094329788697910322018116643, −3.34690679693537797725218067338, −1.86963357379803968775834089507, −1.37376004772787378708517033569, 0.807471010473998115627558476799, 1.93971141854294712875446887944, 3.11638550339419711036394677375, 3.61484060761525062767323608185, 5.14734842874733026831617122001, 5.58434914934996287902734982945, 6.23457711070070281119245033374, 7.28839330378245662837567462962, 7.79908926323512834979920035249, 8.845519500329720016065614957076

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