Properties

Label 2-1440-20.7-c1-0-13
Degree $2$
Conductor $1440$
Sign $0.850 + 0.525i$
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 + (−3 + 3i)7-s + 2i·11-s + (3 − 3i)13-s + (−1 − i)17-s + 4·19-s + (1 + i)23-s + (−3 + 4i)25-s − 10i·31-s + (9 + 3i)35-s + (−1 − i)37-s + 10·41-s + (5 + 5i)43-s + (3 − 3i)47-s − 11i·49-s + ⋯
L(s)  = 1  + (−0.447 − 0.894i)5-s + (−1.13 + 1.13i)7-s + 0.603i·11-s + (0.832 − 0.832i)13-s + (−0.242 − 0.242i)17-s + 0.917·19-s + (0.208 + 0.208i)23-s + (−0.600 + 0.800i)25-s − 1.79i·31-s + (1.52 + 0.507i)35-s + (−0.164 − 0.164i)37-s + 1.56·41-s + (0.762 + 0.762i)43-s + (0.437 − 0.437i)47-s − 1.57i·49-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.850 + 0.525i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.850 + 0.525i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.850 + 0.525i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.287618625\)
\(L(\frac12)\) \(\approx\) \(1.287618625\)
\(L(\frac{3}{2})\) 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 \( 1 + (3 - 3i)T - 7iT^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
13 \( 1 + (-3 + 3i)T - 13iT^{2} \)
17 \( 1 + (1 + i)T + 17iT^{2} \)
19 \( 1 - 4T + 19T^{2} \)
23 \( 1 + (-1 - i)T + 23iT^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 10iT - 31T^{2} \)
37 \( 1 + (1 + i)T + 37iT^{2} \)
41 \( 1 - 10T + 41T^{2} \)
43 \( 1 + (-5 - 5i)T + 43iT^{2} \)
47 \( 1 + (-3 + 3i)T - 47iT^{2} \)
53 \( 1 + (-5 + 5i)T - 53iT^{2} \)
59 \( 1 - 12T + 59T^{2} \)
61 \( 1 - 2T + 61T^{2} \)
67 \( 1 + (1 - i)T - 67iT^{2} \)
71 \( 1 - 2iT - 71T^{2} \)
73 \( 1 + (-1 + i)T - 73iT^{2} \)
79 \( 1 - 8T + 79T^{2} \)
83 \( 1 + (5 + 5i)T + 83iT^{2} \)
89 \( 1 + 16iT - 89T^{2} \)
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

−9.419290581031736656185909176665, −8.768206324466987590914411901827, −7.943813942401990882893960677230, −7.10455879571962736930031763066, −5.87893304226517600485372291989, −5.54327228719001466219278672796, −4.33096906362953271837171940819, −3.38279613115516373263123655981, −2.34531910175480301220125734268, −0.70681604293710395794722326455, 0.943667154249057500058320965285, 2.75114167394878521333379041703, 3.63043030763499192110220570353, 4.13865244319710575234868419509, 5.64856297424818726926114204127, 6.65867682893380635503472243121, 6.92911308460874955430353822287, 7.85831531453200871147971389693, 8.877231507493305594451770038426, 9.634862609776960870229956953185

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