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Properties

Label 2-1440-40.29-c1-0-18
Degree $2$
Conductor $1440$
Sign $0.208 + 0.978i$
Analytic cond. $11.4984$
Root an. cond. $3.39093$
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  + (−0.254 + 2.22i)5-s − 2.64i·7-s + 1.51i·11-s − 3.87·13-s − 3.31i·17-s − 7.08i·19-s − 4.82i·23-s + (−4.87 − 1.12i)25-s + 2.18i·29-s + 7.36·31-s + (5.87 + 0.671i)35-s + 7.87·37-s − 8.72·41-s + 1.01·43-s − 7.08i·47-s + ⋯
L(s)  = 1  + (−0.113 + 0.993i)5-s − 0.998i·7-s + 0.456i·11-s − 1.07·13-s − 0.803i·17-s − 1.62i·19-s − 1.00i·23-s + (−0.974 − 0.225i)25-s + 0.405i·29-s + 1.32·31-s + (0.992 + 0.113i)35-s + 1.29·37-s − 1.36·41-s + 0.155·43-s − 1.03i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.208 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.208 + 0.978i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.208 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.145296396\)
\(L(\frac12)\) \(\approx\) \(1.145296396\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.254 - 2.22i)T \)
good7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 - 1.51iT - 11T^{2} \)
13 \( 1 + 3.87T + 13T^{2} \)
17 \( 1 + 3.31iT - 17T^{2} \)
19 \( 1 + 7.08iT - 19T^{2} \)
23 \( 1 + 4.82iT - 23T^{2} \)
29 \( 1 - 2.18iT - 29T^{2} \)
31 \( 1 - 7.36T + 31T^{2} \)
37 \( 1 - 7.87T + 37T^{2} \)
41 \( 1 + 8.72T + 41T^{2} \)
43 \( 1 - 1.01T + 43T^{2} \)
47 \( 1 + 7.08iT - 47T^{2} \)
53 \( 1 - 4.50T + 53T^{2} \)
59 \( 1 + 6.79iT - 59T^{2} \)
61 \( 1 - 3.60iT - 61T^{2} \)
67 \( 1 + 1.01T + 67T^{2} \)
71 \( 1 + 6.72T + 71T^{2} \)
73 \( 1 + 15.5iT - 73T^{2} \)
79 \( 1 + 7.36T + 79T^{2} \)
83 \( 1 + 7.74T + 83T^{2} \)
89 \( 1 - 14.7T + 89T^{2} \)
97 \( 1 - 11.1iT - 97T^{2} \)
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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

−9.598006825073389683619341664308, −8.505822490351785413777039959246, −7.37910693611239637989634220082, −7.10845411109228499737306126856, −6.36687626106373412795938391870, −4.93187504405301795177736969332, −4.36621179587235101648909897487, −3.08910872355648780256942862807, −2.34562142632008020613831590525, −0.47283714437591873599245760091, 1.36415168842921707948610121588, 2.52297980305578562602860692095, 3.77516929866501466581834916631, 4.72847000613131669972741287511, 5.66017372727742139583128225665, 6.10826274101429481179703461261, 7.54158486927893548345751766520, 8.202351023944800941178376116333, 8.803623546748328352464049232614, 9.708312132273303287030858910208

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