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Properties

Label 2-1440-120.77-c1-0-3
Degree $2$
Conductor $1440$
Sign $-0.925 - 0.379i$
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  + (1.29 + 1.82i)5-s + (0.306 + 0.306i)7-s − 4.06·11-s + (−0.625 − 0.625i)13-s + (−3.57 + 3.57i)17-s − 6.82·19-s + (−1.58 − 1.58i)23-s + (−1.65 + 4.71i)25-s + 8.50i·29-s + 2.56·31-s + (−0.163 + 0.955i)35-s + (1.69 − 1.69i)37-s − 5.19i·41-s + (−4.18 − 4.18i)43-s + (4.58 − 4.58i)47-s + ⋯
L(s)  = 1  + (0.578 + 0.815i)5-s + (0.115 + 0.115i)7-s − 1.22·11-s + (−0.173 − 0.173i)13-s + (−0.867 + 0.867i)17-s − 1.56·19-s + (−0.331 − 0.331i)23-s + (−0.331 + 0.943i)25-s + 1.58i·29-s + 0.461·31-s + (−0.0275 + 0.161i)35-s + (0.277 − 0.277i)37-s − 0.811i·41-s + (−0.638 − 0.638i)43-s + (0.668 − 0.668i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.925 - 0.379i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.925 - 0.379i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (17, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.925 - 0.379i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7006799370\)
\(L(\frac12)\) \(\approx\) \(0.7006799370\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.29 - 1.82i)T \)
good7 \( 1 + (-0.306 - 0.306i)T + 7iT^{2} \)
11 \( 1 + 4.06T + 11T^{2} \)
13 \( 1 + (0.625 + 0.625i)T + 13iT^{2} \)
17 \( 1 + (3.57 - 3.57i)T - 17iT^{2} \)
19 \( 1 + 6.82T + 19T^{2} \)
23 \( 1 + (1.58 + 1.58i)T + 23iT^{2} \)
29 \( 1 - 8.50iT - 29T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 + (-1.69 + 1.69i)T - 37iT^{2} \)
41 \( 1 + 5.19iT - 41T^{2} \)
43 \( 1 + (4.18 + 4.18i)T + 43iT^{2} \)
47 \( 1 + (-4.58 + 4.58i)T - 47iT^{2} \)
53 \( 1 + (7.41 - 7.41i)T - 53iT^{2} \)
59 \( 1 - 8.79iT - 59T^{2} \)
61 \( 1 + 6.08iT - 61T^{2} \)
67 \( 1 + (6.18 - 6.18i)T - 67iT^{2} \)
71 \( 1 - 14.7iT - 71T^{2} \)
73 \( 1 + (-3.05 + 3.05i)T - 73iT^{2} \)
79 \( 1 + 8.56iT - 79T^{2} \)
83 \( 1 + (5.13 - 5.13i)T - 83iT^{2} \)
89 \( 1 - 9.88T + 89T^{2} \)
97 \( 1 + (-1.75 - 1.75i)T + 97iT^{2} \)
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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

−10.23042434036282709646269017306, −8.948750048593418412408630803194, −8.385667105011138817696142708177, −7.37384234638330908902551219436, −6.60451260108518309250538615266, −5.84554034958697384863639607124, −4.96320980695050115624298886757, −3.85721616522791320819897631822, −2.66020273459181548705949306165, −1.96530618670430467857526885000, 0.25070698643963224581451642223, 1.91196774373759151715084023632, 2.75741954908906482184557058401, 4.41147417696126865400665146610, 4.79587278258603415019541028049, 5.90295582356171304653219569243, 6.57047668423243535451906148483, 7.83682826635907260197916255636, 8.271317420199574887823209334377, 9.283558061648344031343291256868

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