↑

Properties

Label 2-2880-16.5-c1-0-27
Degree $2$
Conductor $2880$
Sign $0.468 + 0.883i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (−0.707 − 0.707i)5-s − 1.69i·7-s + (1.72 + 1.72i)11-s + (0.217 − 0.217i)13-s + 3.27·17-s + (−1.73 + 1.73i)19-s − 2.93i·23-s + 1.00i·25-s + (6.42 − 6.42i)29-s − 7.75·31-s + (−1.20 + 1.20i)35-s + (4.70 + 4.70i)37-s − 4.36i·41-s + (4.30 + 4.30i)43-s − 0.568·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 0.642i·7-s + (0.521 + 0.521i)11-s + (0.0602 − 0.0602i)13-s + 0.795·17-s + (−0.397 + 0.397i)19-s − 0.611i·23-s + 0.200i·25-s + (1.19 − 1.19i)29-s − 1.39·31-s + (−0.203 + 0.203i)35-s + (0.772 + 0.772i)37-s − 0.681i·41-s + (0.656 + 0.656i)43-s − 0.0829·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.468 + 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.468 + 0.883i$
Analytic conductor: \(22.9969\)
Root analytic conductor: \(4.79550\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2880} (2161, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2880,\ (\ :1/2),\ 0.468 + 0.883i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.655877737\)
\(L(\frac12)\) \(\approx\) \(1.655877737\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.69iT - 7T^{2} \)
11 \( 1 + (-1.72 - 1.72i)T + 11iT^{2} \)
13 \( 1 + (-0.217 + 0.217i)T - 13iT^{2} \)
17 \( 1 - 3.27T + 17T^{2} \)
19 \( 1 + (1.73 - 1.73i)T - 19iT^{2} \)
23 \( 1 + 2.93iT - 23T^{2} \)
29 \( 1 + (-6.42 + 6.42i)T - 29iT^{2} \)
31 \( 1 + 7.75T + 31T^{2} \)
37 \( 1 + (-4.70 - 4.70i)T + 37iT^{2} \)
41 \( 1 + 4.36iT - 41T^{2} \)
43 \( 1 + (-4.30 - 4.30i)T + 43iT^{2} \)
47 \( 1 + 0.568T + 47T^{2} \)
53 \( 1 + (0.749 + 0.749i)T + 53iT^{2} \)
59 \( 1 + (2.21 + 2.21i)T + 59iT^{2} \)
61 \( 1 + (-7.39 + 7.39i)T - 61iT^{2} \)
67 \( 1 + (9.00 - 9.00i)T - 67iT^{2} \)
71 \( 1 + 2.32iT - 71T^{2} \)
73 \( 1 + 0.285iT - 73T^{2} \)
79 \( 1 - 2.59T + 79T^{2} \)
83 \( 1 + (-10.7 + 10.7i)T - 83iT^{2} \)
89 \( 1 + 10.1iT - 89T^{2} \)
97 \( 1 + 10.0T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.547693244554781480675278982752, −7.88690579630485906986195117306, −7.21350520265028504583846798573, −6.40555655849477979824047356302, −5.59473522962917621064233091077, −4.50351413588767299014888741346, −4.06637057947504275925252697419, −3.02604893301840846166868020099, −1.77219026491112822916367803806, −0.63378914160231156624335469210, 1.07057805934497821207301240950, 2.36969566803137477683204355629, 3.29261028529924888054879632529, 4.04048840826113789049805300520, 5.15339277496242958954467917096, 5.82993818529545198368996152919, 6.63742074975076625487743418971, 7.40094886313014111287397669198, 8.168675483299003488367129210941, 8.987412302631663352987717190245

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