Properties

Label 2-1440-20.7-c1-0-1
Degree $2$
Conductor $1440$
Sign $-0.995 + 0.0898i$
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  + (−1 + 2i)5-s + (−1 + i)7-s + 6i·11-s + (−1 + i)13-s + (−1 − i)17-s + 4·19-s + (−5 − 5i)23-s + (−3 − 4i)25-s − 8i·29-s + 2i·31-s + (−1 − 3i)35-s + (−5 − 5i)37-s − 6·41-s + (3 + 3i)43-s + (−7 + 7i)47-s + ⋯
L(s)  = 1  + (−0.447 + 0.894i)5-s + (−0.377 + 0.377i)7-s + 1.80i·11-s + (−0.277 + 0.277i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s + (−1.04 − 1.04i)23-s + (−0.600 − 0.800i)25-s − 1.48i·29-s + 0.359i·31-s + (−0.169 − 0.507i)35-s + (−0.821 − 0.821i)37-s − 0.937·41-s + (0.457 + 0.457i)43-s + (−1.02 + 1.02i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.995 + 0.0898i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.995 + 0.0898i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (127, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.995 + 0.0898i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5432764664\)
\(L(\frac12)\) \(\approx\) \(0.5432764664\)
\(L(\frac{3}{2})\) 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 \( 1 + (1 - 2i)T \)
good7 \( 1 + (1 - i)T - 7iT^{2} \)
11 \( 1 - 6iT - 11T^{2} \)
13 \( 1 + (1 - i)T - 13iT^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (5 + 5i)T + 23iT^{2} \)
29 \( 1 + 8iT - 29T^{2} \)
31 \( 1 - 2iT - 31T^{2} \)
37 \( 1 + (5 + 5i)T + 37iT^{2} \)
41 \( 1 + 6T + 41T^{2} \)
43 \( 1 + (-3 - 3i)T + 43iT^{2} \)
47 \( 1 + (7 - 7i)T - 47iT^{2} \)
53 \( 1 + (-1 + i)T - 53iT^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (7 - 7i)T - 67iT^{2} \)
71 \( 1 - 6iT - 71T^{2} \)
73 \( 1 + (-9 + 9i)T - 73iT^{2} \)
79 \( 1 + 8T + 79T^{2} \)
83 \( 1 + (-5 - 5i)T + 83iT^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 + (3 + 3i)T + 97iT^{2} \)
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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

−9.869073141952054986676514834988, −9.406806471027853192097028606626, −8.173110521573455575441686732743, −7.42440428137278816054368060778, −6.80957139102874314198658143881, −6.01032625509004315312111165013, −4.76682241780314307731601842257, −4.04282229855996479363070751214, −2.83197027244221292361290427993, −2.02072749519473538072721194102, 0.21634035022507637690139021622, 1.46011209873107107542489244705, 3.27878314372279031627482886553, 3.72118649948981497388303662435, 5.07201701853113120937285396111, 5.63581219000486920348424013005, 6.67619422419343276746333024759, 7.66936865878311624725629441079, 8.364648117526184691794388783482, 8.982057308331074596731283697574

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