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Properties

Label 2-2880-48.11-c1-0-8
Degree $2$
Conductor $2880$
Sign $0.803 - 0.594i$
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 + 0.296·7-s + (−4.14 − 4.14i)11-s + (−3.69 + 3.69i)13-s − 5.57i·17-s + (2.26 + 2.26i)19-s + 0.797i·23-s − 1.00i·25-s + (6.66 + 6.66i)29-s − 2.04i·31-s + (−0.209 + 0.209i)35-s + (2.62 + 2.62i)37-s + 9.97·41-s + (−2.77 + 2.77i)43-s + 5.51·47-s + ⋯
L(s)  = 1  + (−0.316 + 0.316i)5-s + 0.112·7-s + (−1.24 − 1.24i)11-s + (−1.02 + 1.02i)13-s − 1.35i·17-s + (0.518 + 0.518i)19-s + 0.166i·23-s − 0.200i·25-s + (1.23 + 1.23i)29-s − 0.367i·31-s + (−0.0354 + 0.0354i)35-s + (0.431 + 0.431i)37-s + 1.55·41-s + (−0.423 + 0.423i)43-s + 0.805·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.803 - 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.803 - 0.594i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (1871, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.803 - 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.313170637\)
\(L(\frac12)\) \(\approx\) \(1.313170637\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.296T + 7T^{2} \)
11 \( 1 + (4.14 + 4.14i)T + 11iT^{2} \)
13 \( 1 + (3.69 - 3.69i)T - 13iT^{2} \)
17 \( 1 + 5.57iT - 17T^{2} \)
19 \( 1 + (-2.26 - 2.26i)T + 19iT^{2} \)
23 \( 1 - 0.797iT - 23T^{2} \)
29 \( 1 + (-6.66 - 6.66i)T + 29iT^{2} \)
31 \( 1 + 2.04iT - 31T^{2} \)
37 \( 1 + (-2.62 - 2.62i)T + 37iT^{2} \)
41 \( 1 - 9.97T + 41T^{2} \)
43 \( 1 + (2.77 - 2.77i)T - 43iT^{2} \)
47 \( 1 - 5.51T + 47T^{2} \)
53 \( 1 + (-7.43 + 7.43i)T - 53iT^{2} \)
59 \( 1 + (-9.43 - 9.43i)T + 59iT^{2} \)
61 \( 1 + (-0.274 + 0.274i)T - 61iT^{2} \)
67 \( 1 + (-8.79 - 8.79i)T + 67iT^{2} \)
71 \( 1 + 0.256iT - 71T^{2} \)
73 \( 1 - 1.18iT - 73T^{2} \)
79 \( 1 - 10.9iT - 79T^{2} \)
83 \( 1 + (10.1 - 10.1i)T - 83iT^{2} \)
89 \( 1 - 2.04T + 89T^{2} \)
97 \( 1 - 0.781T + 97T^{2} \)
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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

−8.742716974529381942338062746532, −8.083633265911438750798728785749, −7.31036489954839364443192449643, −6.76968482106029316941020908592, −5.62918972055008088415500711230, −5.06801655002049588925060064311, −4.14110626627077817492242818235, −2.99172988539732721705860394278, −2.49271066503337836687578883990, −0.832915822746408396837234876305, 0.56830280764726611539437561502, 2.13080972898375319402610584004, 2.84185038226105806005274735562, 4.10091754078622937350458266237, 4.84266513844135272019114152715, 5.43853459068976558667345635242, 6.42506727678207756452894944094, 7.55899420859570069647813638337, 7.73166025562317579166484210594, 8.554229566560677570971205593325

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