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Properties

Label 2-1440-40.27-c1-0-23
Degree $2$
Conductor $1440$
Sign $-0.249 + 0.968i$
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  + (0.386 + 2.20i)5-s + (1.51 − 1.51i)7-s − 3.92·11-s + (−3.56 − 3.56i)13-s + (−1.37 − 1.37i)17-s − 4i·19-s + (−5.17 − 5.17i)23-s + (−4.70 + 1.70i)25-s + 5.95·29-s − 7.12i·31-s + (3.92 + 2.75i)35-s + (3.56 − 3.56i)37-s + 2.75·41-s + (−5.40 + 5.40i)43-s + (−1.54 + 1.54i)47-s + ⋯
L(s)  = 1  + (0.172 + 0.984i)5-s + (0.573 − 0.573i)7-s − 1.18·11-s + (−0.988 − 0.988i)13-s + (−0.333 − 0.333i)17-s − 0.917i·19-s + (−1.07 − 1.07i)23-s + (−0.940 + 0.340i)25-s + 1.10·29-s − 1.28i·31-s + (0.663 + 0.465i)35-s + (0.585 − 0.585i)37-s + 0.430·41-s + (−0.823 + 0.823i)43-s + (−0.225 + 0.225i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.249 + 0.968i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (847, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.249 + 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.9233473658\)
\(L(\frac12)\) \(\approx\) \(0.9233473658\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.386 - 2.20i)T \)
good7 \( 1 + (-1.51 + 1.51i)T - 7iT^{2} \)
11 \( 1 + 3.92T + 11T^{2} \)
13 \( 1 + (3.56 + 3.56i)T + 13iT^{2} \)
17 \( 1 + (1.37 + 1.37i)T + 17iT^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + (5.17 + 5.17i)T + 23iT^{2} \)
29 \( 1 - 5.95T + 29T^{2} \)
31 \( 1 + 7.12iT - 31T^{2} \)
37 \( 1 + (-3.56 + 3.56i)T - 37iT^{2} \)
41 \( 1 - 2.75T + 41T^{2} \)
43 \( 1 + (5.40 - 5.40i)T - 43iT^{2} \)
47 \( 1 + (1.54 - 1.54i)T - 47iT^{2} \)
53 \( 1 + (-1.81 - 1.81i)T + 53iT^{2} \)
59 \( 1 - 3.92iT - 59T^{2} \)
61 \( 1 + 13.1iT - 61T^{2} \)
67 \( 1 + (-5.40 - 5.40i)T + 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 + 5i)T - 73iT^{2} \)
79 \( 1 + 7.12T + 79T^{2} \)
83 \( 1 + (-6.67 + 6.67i)T - 83iT^{2} \)
89 \( 1 - 18.4iT - 89T^{2} \)
97 \( 1 + (10.4 + 10.4i)T + 97iT^{2} \)
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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

−9.569544381245730361136726983767, −8.136815798313415852042499659667, −7.77795432961411919579482148273, −6.95709632201700849029948456759, −6.05655182760305497848128685089, −5.05668688938232821006883462256, −4.28519336140794073278167544531, −2.85223679329530172869443597504, −2.36082121249390053181332182039, −0.34698177556116503026306615818, 1.59640466971338490118921677845, 2.43717745223481739602699418932, 3.94013527038445357995590528157, 4.95309343648326947738995276655, 5.36163084272325602989404574965, 6.40693107221520776968547541149, 7.56340250547870497196341923424, 8.253792401057628784475804549139, 8.820949904164934707374710426805, 9.831311907098428108348441357118

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