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Properties

Label 2-1440-120.53-c1-0-21
Degree $2$
Conductor $1440$
Sign $-0.791 + 0.611i$
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  + (−2.03 − 0.929i)5-s + (2.49 − 2.49i)7-s − 3.92·11-s + (4.55 − 4.55i)13-s + (1.88 + 1.88i)17-s − 4.61·19-s + (0.741 − 0.741i)23-s + (3.27 + 3.78i)25-s + 4.35i·29-s − 9.67·31-s + (−7.39 + 2.75i)35-s + (−5.39 − 5.39i)37-s − 6.33i·41-s + (−0.206 + 0.206i)43-s + (−3.48 − 3.48i)47-s + ⋯
L(s)  = 1  + (−0.909 − 0.415i)5-s + (0.942 − 0.942i)7-s − 1.18·11-s + (1.26 − 1.26i)13-s + (0.457 + 0.457i)17-s − 1.05·19-s + (0.154 − 0.154i)23-s + (0.654 + 0.756i)25-s + 0.809i·29-s − 1.73·31-s + (−1.24 + 0.465i)35-s + (−0.887 − 0.887i)37-s − 0.989i·41-s + (−0.0314 + 0.0314i)43-s + (−0.507 − 0.507i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.791 + 0.611i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.791 + 0.611i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (593, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.791 + 0.611i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9165936182\)
\(L(\frac12)\) \(\approx\) \(0.9165936182\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.03 + 0.929i)T \)
good7 \( 1 + (-2.49 + 2.49i)T - 7iT^{2} \)
11 \( 1 + 3.92T + 11T^{2} \)
13 \( 1 + (-4.55 + 4.55i)T - 13iT^{2} \)
17 \( 1 + (-1.88 - 1.88i)T + 17iT^{2} \)
19 \( 1 + 4.61T + 19T^{2} \)
23 \( 1 + (-0.741 + 0.741i)T - 23iT^{2} \)
29 \( 1 - 4.35iT - 29T^{2} \)
31 \( 1 + 9.67T + 31T^{2} \)
37 \( 1 + (5.39 + 5.39i)T + 37iT^{2} \)
41 \( 1 + 6.33iT - 41T^{2} \)
43 \( 1 + (0.206 - 0.206i)T - 43iT^{2} \)
47 \( 1 + (3.48 + 3.48i)T + 47iT^{2} \)
53 \( 1 + (1.01 + 1.01i)T + 53iT^{2} \)
59 \( 1 + 0.531iT - 59T^{2} \)
61 \( 1 + 3.00iT - 61T^{2} \)
67 \( 1 + (-1.28 - 1.28i)T + 67iT^{2} \)
71 \( 1 + 7.61iT - 71T^{2} \)
73 \( 1 + (0.509 + 0.509i)T + 73iT^{2} \)
79 \( 1 + 1.31iT - 79T^{2} \)
83 \( 1 + (9.85 + 9.85i)T + 83iT^{2} \)
89 \( 1 - 2.91T + 89T^{2} \)
97 \( 1 + (8.11 - 8.11i)T - 97iT^{2} \)
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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

−8.895164028398884515450242764616, −8.274745834506209230545842214549, −7.76327468177152641760279987441, −7.05706486306698391299831194454, −5.66850018582559303176427548903, −5.03779116210358753902261508915, −3.99084175558494955120807718584, −3.34056582580329436601272074559, −1.66487807034717919465243596391, −0.36889346683614037308638796940, 1.71321900482583369585822176370, 2.79178270218068990300483589236, 3.91586777256306658637973310335, 4.78695556391444417338821433401, 5.68473627117091360125245755337, 6.64158812352600736490642270725, 7.55719831303431479569122202429, 8.354794200182943849615610061223, 8.722682081837264657380533508742, 9.842727594582499721768562279140

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