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Properties

Label 2-2880-16.5-c1-0-12
Degree $2$
Conductor $2880$
Sign $0.849 - 0.526i$
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 − 0.690i·7-s + (−3.06 − 3.06i)11-s + (−2.33 + 2.33i)13-s + 5.28·17-s + (−5.38 + 5.38i)19-s + 1.60i·23-s + 1.00i·25-s + (−1.70 + 1.70i)29-s + 4.69·31-s + (−0.488 + 0.488i)35-s + (7.89 + 7.89i)37-s − 5.49i·41-s + (0.256 + 0.256i)43-s − 4.60·47-s + ⋯
L(s)  = 1  + (−0.316 − 0.316i)5-s − 0.261i·7-s + (−0.922 − 0.922i)11-s + (−0.648 + 0.648i)13-s + 1.28·17-s + (−1.23 + 1.23i)19-s + 0.335i·23-s + 0.200i·25-s + (−0.316 + 0.316i)29-s + 0.843·31-s + (−0.0825 + 0.0825i)35-s + (1.29 + 1.29i)37-s − 0.858i·41-s + (0.0390 + 0.0390i)43-s − 0.672·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.849 - 0.526i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2880\)    =    \(2^{6} \cdot 3^{2} \cdot 5\)
Sign: $0.849 - 0.526i$
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: \((2,\ 2880,\ (\ :1/2),\ 0.849 - 0.526i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.304239534\)
\(L(\frac12)\) \(\approx\) \(1.304239534\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.690iT - 7T^{2} \)
11 \( 1 + (3.06 + 3.06i)T + 11iT^{2} \)
13 \( 1 + (2.33 - 2.33i)T - 13iT^{2} \)
17 \( 1 - 5.28T + 17T^{2} \)
19 \( 1 + (5.38 - 5.38i)T - 19iT^{2} \)
23 \( 1 - 1.60iT - 23T^{2} \)
29 \( 1 + (1.70 - 1.70i)T - 29iT^{2} \)
31 \( 1 - 4.69T + 31T^{2} \)
37 \( 1 + (-7.89 - 7.89i)T + 37iT^{2} \)
41 \( 1 + 5.49iT - 41T^{2} \)
43 \( 1 + (-0.256 - 0.256i)T + 43iT^{2} \)
47 \( 1 + 4.60T + 47T^{2} \)
53 \( 1 + (-4.99 - 4.99i)T + 53iT^{2} \)
59 \( 1 + (-1.46 - 1.46i)T + 59iT^{2} \)
61 \( 1 + (-9.33 + 9.33i)T - 61iT^{2} \)
67 \( 1 + (-1.94 + 1.94i)T - 67iT^{2} \)
71 \( 1 - 2.32iT - 71T^{2} \)
73 \( 1 - 1.29iT - 73T^{2} \)
79 \( 1 - 5.01T + 79T^{2} \)
83 \( 1 + (-7.30 + 7.30i)T - 83iT^{2} \)
89 \( 1 - 1.81iT - 89T^{2} \)
97 \( 1 - 5.27T + 97T^{2} \)
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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

−8.661354336094488925903704650381, −8.048656428530063348859663278013, −7.56187813054275007050702547076, −6.50892516609212484070034529203, −5.74288749940959465793176541401, −4.98763107918865951639576518449, −4.06704808736480030888515596288, −3.27739554105944551845129828612, −2.19121917343802885298490127849, −0.886867924974046900431668741992, 0.53661699123013134077951861291, 2.30658656836745721965959495414, 2.77623866771371448036050570108, 4.02091297354587641539665922163, 4.84313853657822459708140488543, 5.54364270453652507304649281349, 6.49762514637040775831235179192, 7.34448959329266589450105896696, 7.84989941785897482862163929710, 8.580259683829034008799188389929

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