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Properties

Label 2-1440-120.77-c1-0-0
Degree $2$
Conductor $1440$
Sign $-0.494 - 0.869i$
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  + (−1.42 − 1.72i)5-s + (−3.11 − 3.11i)7-s − 1.17·11-s + (−2.15 − 2.15i)13-s + (1.33 − 1.33i)17-s + 0.322·19-s + (4.71 + 4.71i)23-s + (−0.933 + 4.91i)25-s + 6.63i·29-s + 0.0675·31-s + (−0.924 + 9.82i)35-s + (−7.60 + 7.60i)37-s + 3.19i·41-s + (−6.70 − 6.70i)43-s + (7.34 − 7.34i)47-s + ⋯
L(s)  = 1  + (−0.637 − 0.770i)5-s + (−1.17 − 1.17i)7-s − 0.355·11-s + (−0.596 − 0.596i)13-s + (0.323 − 0.323i)17-s + 0.0739·19-s + (0.982 + 0.982i)23-s + (−0.186 + 0.982i)25-s + 1.23i·29-s + 0.0121·31-s + (−0.156 + 1.66i)35-s + (−1.25 + 1.25i)37-s + 0.499i·41-s + (−1.02 − 1.02i)43-s + (1.07 − 1.07i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.494 - 0.869i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.494 - 0.869i$
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: \((2,\ 1440,\ (\ :1/2),\ -0.494 - 0.869i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03351133512\)
\(L(\frac12)\) \(\approx\) \(0.03351133512\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.42 + 1.72i)T \)
good7 \( 1 + (3.11 + 3.11i)T + 7iT^{2} \)
11 \( 1 + 1.17T + 11T^{2} \)
13 \( 1 + (2.15 + 2.15i)T + 13iT^{2} \)
17 \( 1 + (-1.33 + 1.33i)T - 17iT^{2} \)
19 \( 1 - 0.322T + 19T^{2} \)
23 \( 1 + (-4.71 - 4.71i)T + 23iT^{2} \)
29 \( 1 - 6.63iT - 29T^{2} \)
31 \( 1 - 0.0675T + 31T^{2} \)
37 \( 1 + (7.60 - 7.60i)T - 37iT^{2} \)
41 \( 1 - 3.19iT - 41T^{2} \)
43 \( 1 + (6.70 + 6.70i)T + 43iT^{2} \)
47 \( 1 + (-7.34 + 7.34i)T - 47iT^{2} \)
53 \( 1 + (5.73 - 5.73i)T - 53iT^{2} \)
59 \( 1 - 8.68iT - 59T^{2} \)
61 \( 1 + 12.5iT - 61T^{2} \)
67 \( 1 + (-1.87 + 1.87i)T - 67iT^{2} \)
71 \( 1 - 4.18iT - 71T^{2} \)
73 \( 1 + (-3.97 + 3.97i)T - 73iT^{2} \)
79 \( 1 - 9.66iT - 79T^{2} \)
83 \( 1 + (0.585 - 0.585i)T - 83iT^{2} \)
89 \( 1 + 0.557T + 89T^{2} \)
97 \( 1 + (10.5 + 10.5i)T + 97iT^{2} \)
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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

−9.831452585321228362897184426712, −9.048165916004287401518260858563, −8.136618121226817568676985222509, −7.25319314654531477817098839727, −6.87891498779655710922241106408, −5.46123000628701447896692118211, −4.82643875791271423717716342504, −3.64533506621670294320469604327, −3.11101085415838639798800458681, −1.17886982765473592147198342943, 0.01462141945389912985469192836, 2.33147226180699236931122597242, 2.98885445751593124910792540232, 3.97645846747272166697616185320, 5.14373960283042806920844534580, 6.15212057154824883632522932706, 6.73834773240118500914830027634, 7.58485584232525329270149991332, 8.522533807125648952698670771734, 9.291683008902134484098942548046

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