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Properties

Label 2-2880-15.8-c1-0-38
Degree $2$
Conductor $2880$
Sign $0.374 + 0.927i$
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 + (3.46 − 3.46i)7-s + 4.89i·11-s + (−1 − i)13-s + (−2.82 − 2.82i)17-s − 6.92i·19-s + (3.99 − 3i)25-s + 7.07·29-s − 6.92·31-s + (4.89 − 9.79i)35-s + (−1 + i)37-s − 9.89i·41-s + (6.92 + 6.92i)43-s + (4.89 + 4.89i)47-s − 16.9i·49-s + ⋯
L(s)  = 1  + (0.948 − 0.316i)5-s + (1.30 − 1.30i)7-s + 1.47i·11-s + (−0.277 − 0.277i)13-s + (−0.685 − 0.685i)17-s − 1.58i·19-s + (0.799 − 0.600i)25-s + 1.31·29-s − 1.24·31-s + (0.828 − 1.65i)35-s + (−0.164 + 0.164i)37-s − 1.54i·41-s + (1.05 + 1.05i)43-s + (0.714 + 0.714i)47-s − 2.42i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.374 + 0.927i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.374 + 0.927i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.374 + 0.927i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.482632782\)
\(L(\frac12)\) \(\approx\) \(2.482632782\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.46 + 3.46i)T - 7iT^{2} \)
11 \( 1 - 4.89iT - 11T^{2} \)
13 \( 1 + (1 + i)T + 13iT^{2} \)
17 \( 1 + (2.82 + 2.82i)T + 17iT^{2} \)
19 \( 1 + 6.92iT - 19T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 - 7.07T + 29T^{2} \)
31 \( 1 + 6.92T + 31T^{2} \)
37 \( 1 + (1 - i)T - 37iT^{2} \)
41 \( 1 + 9.89iT - 41T^{2} \)
43 \( 1 + (-6.92 - 6.92i)T + 43iT^{2} \)
47 \( 1 + (-4.89 - 4.89i)T + 47iT^{2} \)
53 \( 1 + (-2.82 + 2.82i)T - 53iT^{2} \)
59 \( 1 + 4.89T + 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + (6.92 - 6.92i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (7 + 7i)T + 73iT^{2} \)
79 \( 1 + 6.92iT - 79T^{2} \)
83 \( 1 + (4.89 - 4.89i)T - 83iT^{2} \)
89 \( 1 - 9.89T + 89T^{2} \)
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

−8.835628323074854879986794612401, −7.59132156109862461998863768815, −7.28329558081162477115113811128, −6.52394699571631571567864004318, −5.24610100123623684280698150023, −4.71639154279443507275098139552, −4.26397581790127510267626740225, −2.64033539227939357913588643296, −1.85051261872529498005956676953, −0.805249522424721214980345985212, 1.45007702003448595143497928063, 2.17821777130952702205467232807, 3.08500776460302401355617631349, 4.27200466570526603989060694747, 5.33124051878277486638329489194, 5.81670449852724423433205172878, 6.32671201696437225158137890874, 7.52989412575774616468006930182, 8.464075904383881584141105791969, 8.692712066536951152404951694123

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