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Properties

Label 2-2880-16.13-c1-0-10
Degree $2$
Conductor $2880$
Sign $-0.659 - 0.751i$
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 + 4.02i·7-s + (−0.646 + 0.646i)11-s + (4.91 + 4.91i)13-s + 2.70·17-s + (0.438 + 0.438i)19-s + 3.60i·23-s − 1.00i·25-s + (−2.00 − 2.00i)29-s − 4.30·31-s + (−2.84 − 2.84i)35-s + (−0.743 + 0.743i)37-s + 0.603i·41-s + (5.03 − 5.03i)43-s + 10.8·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + 1.52i·7-s + (−0.195 + 0.195i)11-s + (1.36 + 1.36i)13-s + 0.656·17-s + (0.100 + 0.100i)19-s + 0.750i·23-s − 0.200i·25-s + (−0.373 − 0.373i)29-s − 0.774·31-s + (−0.481 − 0.481i)35-s + (−0.122 + 0.122i)37-s + 0.0943i·41-s + (0.767 − 0.767i)43-s + 1.57·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.659 - 0.751i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $-0.659 - 0.751i$
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: \((2,\ 2880,\ (\ :1/2),\ -0.659 - 0.751i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.535918867\)
\(L(\frac12)\) \(\approx\) \(1.535918867\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.02iT - 7T^{2} \)
11 \( 1 + (0.646 - 0.646i)T - 11iT^{2} \)
13 \( 1 + (-4.91 - 4.91i)T + 13iT^{2} \)
17 \( 1 - 2.70T + 17T^{2} \)
19 \( 1 + (-0.438 - 0.438i)T + 19iT^{2} \)
23 \( 1 - 3.60iT - 23T^{2} \)
29 \( 1 + (2.00 + 2.00i)T + 29iT^{2} \)
31 \( 1 + 4.30T + 31T^{2} \)
37 \( 1 + (0.743 - 0.743i)T - 37iT^{2} \)
41 \( 1 - 0.603iT - 41T^{2} \)
43 \( 1 + (-5.03 + 5.03i)T - 43iT^{2} \)
47 \( 1 - 10.8T + 47T^{2} \)
53 \( 1 + (4.07 - 4.07i)T - 53iT^{2} \)
59 \( 1 + (-1.22 + 1.22i)T - 59iT^{2} \)
61 \( 1 + (6.98 + 6.98i)T + 61iT^{2} \)
67 \( 1 + (5.24 + 5.24i)T + 67iT^{2} \)
71 \( 1 - 13.7iT - 71T^{2} \)
73 \( 1 + 1.30iT - 73T^{2} \)
79 \( 1 + 0.611T + 79T^{2} \)
83 \( 1 + (-1.29 - 1.29i)T + 83iT^{2} \)
89 \( 1 - 10.9iT - 89T^{2} \)
97 \( 1 + 12.7T + 97T^{2} \)
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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

−9.159776199828504641466023860206, −8.361849760500393826296445208136, −7.59734598431589432959338467309, −6.71956856540174982122116880148, −5.89826971462876759550643730696, −5.42510201482606007858162144290, −4.22190021342095894519507797605, −3.46818932225154102577114527915, −2.43423251355292341613341908564, −1.52430991053254894532299044337, 0.53538844710510925036933647822, 1.32388391976543762356404584582, 3.02154418956695063551204959732, 3.71825955801444165473167277776, 4.42290115934927625457144089522, 5.46838117231996121577450307760, 6.13852387296515953594931062704, 7.21811055615128267127689478945, 7.70567800105031534434147129927, 8.380100164253408989687936198491

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