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Properties

Label 2-1440-120.53-c1-0-15
Degree $2$
Conductor $1440$
Sign $0.877 + 0.479i$
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  + (2.23 − 0.0113i)5-s + (−0.471 + 0.471i)7-s + 0.335·11-s + (3.50 − 3.50i)13-s + (−2.53 − 2.53i)17-s − 4.07·19-s + (6.20 − 6.20i)23-s + (4.99 − 0.0506i)25-s + 2.42i·29-s + 6.41·31-s + (−1.04 + 1.06i)35-s + (2.24 + 2.24i)37-s − 5.80i·41-s + (−4.87 + 4.87i)43-s + (−1.68 − 1.68i)47-s + ⋯
L(s)  = 1  + (0.999 − 0.00506i)5-s + (−0.178 + 0.178i)7-s + 0.101·11-s + (0.971 − 0.971i)13-s + (−0.614 − 0.614i)17-s − 0.934·19-s + (1.29 − 1.29i)23-s + (0.999 − 0.0101i)25-s + 0.450i·29-s + 1.15·31-s + (−0.177 + 0.179i)35-s + (0.369 + 0.369i)37-s − 0.906i·41-s + (−0.743 + 0.743i)43-s + (−0.245 − 0.245i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.877 + 0.479i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.877 + 0.479i$
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: \((2,\ 1440,\ (\ :1/2),\ 0.877 + 0.479i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.047271705\)
\(L(\frac12)\) \(\approx\) \(2.047271705\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.23 + 0.0113i)T \)
good7 \( 1 + (0.471 - 0.471i)T - 7iT^{2} \)
11 \( 1 - 0.335T + 11T^{2} \)
13 \( 1 + (-3.50 + 3.50i)T - 13iT^{2} \)
17 \( 1 + (2.53 + 2.53i)T + 17iT^{2} \)
19 \( 1 + 4.07T + 19T^{2} \)
23 \( 1 + (-6.20 + 6.20i)T - 23iT^{2} \)
29 \( 1 - 2.42iT - 29T^{2} \)
31 \( 1 - 6.41T + 31T^{2} \)
37 \( 1 + (-2.24 - 2.24i)T + 37iT^{2} \)
41 \( 1 + 5.80iT - 41T^{2} \)
43 \( 1 + (4.87 - 4.87i)T - 43iT^{2} \)
47 \( 1 + (1.68 + 1.68i)T + 47iT^{2} \)
53 \( 1 + (-3.05 - 3.05i)T + 53iT^{2} \)
59 \( 1 + 12.2iT - 59T^{2} \)
61 \( 1 - 7.49iT - 61T^{2} \)
67 \( 1 + (-5.55 - 5.55i)T + 67iT^{2} \)
71 \( 1 + 13.4iT - 71T^{2} \)
73 \( 1 + (-5.05 - 5.05i)T + 73iT^{2} \)
79 \( 1 + 8.85iT - 79T^{2} \)
83 \( 1 + (-4.78 - 4.78i)T + 83iT^{2} \)
89 \( 1 - 8.33T + 89T^{2} \)
97 \( 1 + (-10.1 + 10.1i)T - 97iT^{2} \)
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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

−9.333743365131281296209764850993, −8.790613444842578259620258766785, −8.035548097626450718939408382305, −6.66761102026378866284969717651, −6.37236615708972629288779084079, −5.32285124404160001964490802850, −4.54447578198751415176542673078, −3.16950681264866267996799307182, −2.35366483586298072035356426042, −0.942584217100777250429856358762, 1.31512465168942688441178417200, 2.30947716913307599704037377331, 3.57905363849406877761277908545, 4.52679763278151969220325480492, 5.55199579023269742368257718625, 6.47601905994544712606167386399, 6.81083259867675140734300466109, 8.171326639756669788017576315366, 8.904991805763827429993585500480, 9.515765582710164551790305447043

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