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Properties

Label 2-1440-120.59-c1-0-11
Degree $2$
Conductor $1440$
Sign $0.577 - 0.816i$
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  + 2.23i·5-s + 5.16·7-s + 3.05i·11-s + 0.837·13-s + 6.32·19-s − 4.47i·23-s − 5.00·25-s + 11.5i·35-s − 11.1·37-s + 10.3i·41-s − 2.82i·47-s + 19.6·49-s − 5.65i·53-s − 6.83·55-s − 5.42i·59-s + ⋯
L(s)  = 1  + 0.999i·5-s + 1.95·7-s + 0.921i·11-s + 0.232·13-s + 1.45·19-s − 0.932i·23-s − 1.00·25-s + 1.95i·35-s − 1.83·37-s + 1.61i·41-s − 0.412i·47-s + 2.80·49-s − 0.777i·53-s − 0.921·55-s − 0.706i·59-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.577 - 0.816i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.577 - 0.816i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (719, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.577 - 0.816i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.166431526\)
\(L(\frac12)\) \(\approx\) \(2.166431526\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.23iT \)
good7 \( 1 - 5.16T + 7T^{2} \)
11 \( 1 - 3.05iT - 11T^{2} \)
13 \( 1 - 0.837T + 13T^{2} \)
17 \( 1 + 17T^{2} \)
19 \( 1 - 6.32T + 19T^{2} \)
23 \( 1 + 4.47iT - 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 - 10.3iT - 41T^{2} \)
43 \( 1 - 43T^{2} \)
47 \( 1 + 2.82iT - 47T^{2} \)
53 \( 1 + 5.65iT - 53T^{2} \)
59 \( 1 + 5.42iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 - 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 - 18.8iT - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−9.822769143754019984684795114027, −8.712211722488543257382266493455, −7.912320671900962996779291545946, −7.35276415872876471285679535925, −6.53889078164023175883575719433, −5.31355522898888937138739690189, −4.73527970265223480109049146475, −3.63612186421459995097202438735, −2.39965056854226908410061194493, −1.47105384809912020728087156705, 1.01811167614650883288383877307, 1.84044578635171724083271146122, 3.44756748908849115914978696017, 4.46601847594871413191489388810, 5.34972775812709278694710594823, 5.66002088388362004661863905068, 7.30506864029499603240280840934, 7.86070368209307646014066309646, 8.685236051886864542729471844427, 9.062283524425959024029995913193

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