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Properties

Label 2-1440-120.53-c1-0-20
Degree $2$
Conductor $1440$
Sign $0.840 + 0.541i$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.840 + 0.541i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.840 + 0.541i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.840 + 0.541i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.380525233\)
\(L(\frac12)\) \(\approx\) \(2.380525233\)
\(L(\frac{3}{2})\) 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

−9.425192552862526948776531239605, −8.709570662251242778845186138956, −7.83999040154943044802804044342, −6.96425586616244759815069231415, −6.18971232905169329465050085270, −5.41983181260277321360836825906, −4.26206550818307923561237616528, −3.49580922608865547669316403280, −2.06861080810491473634640150777, −1.07617566062421137725907104084, 1.65478176293444197562739949179, 1.94583959274105495149812734217, 3.71904472586850539334528436245, 4.57597068409972231695687081209, 5.54594170794185043884842809043, 6.27844676736525990716911856817, 6.93072175147327049694359664001, 8.458702677789087914461056945250, 8.857264602950313343570380374981, 9.197741217882916364267328974705

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