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Properties

Label 2-1440-40.3-c1-0-1
Degree $2$
Conductor $1440$
Sign $-0.0346 - 0.999i$
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  + (−1.83 − 1.28i)5-s + (−2.94 − 2.94i)7-s − 1.61·11-s + (−2.50 + 2.50i)13-s + (4.59 − 4.59i)17-s + 4i·19-s + (1.09 − 1.09i)23-s + (1.70 + 4.70i)25-s − 4.75·29-s + 5.01i·31-s + (1.61 + 9.18i)35-s + (2.50 + 2.50i)37-s − 9.18·41-s + (7.40 + 7.40i)43-s + (7.32 + 7.32i)47-s + ⋯
L(s)  = 1  + (−0.818 − 0.574i)5-s + (−1.11 − 1.11i)7-s − 0.485·11-s + (−0.696 + 0.696i)13-s + (1.11 − 1.11i)17-s + 0.917i·19-s + (0.227 − 0.227i)23-s + (0.340 + 0.940i)25-s − 0.882·29-s + 0.901i·31-s + (0.272 + 1.55i)35-s + (0.412 + 0.412i)37-s − 1.43·41-s + (1.12 + 1.12i)43-s + (1.06 + 1.06i)47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0346 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.0346 - 0.999i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1423, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.0346 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.4075080862\)
\(L(\frac12)\) \(\approx\) \(0.4075080862\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.83 + 1.28i)T \)
good7 \( 1 + (2.94 + 2.94i)T + 7iT^{2} \)
11 \( 1 + 1.61T + 11T^{2} \)
13 \( 1 + (2.50 - 2.50i)T - 13iT^{2} \)
17 \( 1 + (-4.59 + 4.59i)T - 17iT^{2} \)
19 \( 1 - 4iT - 19T^{2} \)
23 \( 1 + (-1.09 + 1.09i)T - 23iT^{2} \)
29 \( 1 + 4.75T + 29T^{2} \)
31 \( 1 - 5.01iT - 31T^{2} \)
37 \( 1 + (-2.50 - 2.50i)T + 37iT^{2} \)
41 \( 1 + 9.18T + 41T^{2} \)
43 \( 1 + (-7.40 - 7.40i)T + 43iT^{2} \)
47 \( 1 + (-7.32 - 7.32i)T + 47iT^{2} \)
53 \( 1 + (-3.11 + 3.11i)T - 53iT^{2} \)
59 \( 1 + 1.61iT - 59T^{2} \)
61 \( 1 + 6.78iT - 61T^{2} \)
67 \( 1 + (7.40 - 7.40i)T - 67iT^{2} \)
71 \( 1 - 71T^{2} \)
73 \( 1 + (-5 - 5i)T + 73iT^{2} \)
79 \( 1 + 5.01T + 79T^{2} \)
83 \( 1 + (7.57 + 7.57i)T + 83iT^{2} \)
89 \( 1 - 2.74iT - 89T^{2} \)
97 \( 1 + (-2.40 + 2.40i)T - 97iT^{2} \)
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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

−9.746194652224661311385139263046, −9.067713031167448157047722713868, −7.895821491846796258609208397416, −7.40383724150733225412455588391, −6.70049850123988281053830691651, −5.49650246996514523485138221249, −4.58706127105776080872729329105, −3.74107277702385257728458854405, −2.92134041427551048845159988055, −1.09542874220045409521151806152, 0.18754924294023134121516208702, 2.40230742270763332060547324064, 3.12174003887694708642313807129, 3.98686787337111056406017468302, 5.39836185339553406990684878473, 5.91267118677006311333306804123, 7.01365820926927757855097112175, 7.64259316192041480626717881922, 8.516017782899423498954210144016, 9.328801610762288168527075089067

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