Properties

Label 2-1440-40.29-c1-0-27
Degree $2$
Conductor $1440$
Sign $-0.632 + 0.774i$
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  + (1.41 − 1.73i)5-s − 4.89i·7-s − 3.46i·11-s + (−0.999 − 4.89i)25-s + 10.3i·29-s − 10·31-s + (−8.48 − 6.92i)35-s − 16.9·49-s + 14.1·53-s + (−5.99 − 4.89i)55-s + 10.3i·59-s − 9.79i·73-s − 16.9·77-s + 10·79-s + 5.65·83-s + ⋯
L(s)  = 1  + (0.632 − 0.774i)5-s − 1.85i·7-s − 1.04i·11-s + (−0.199 − 0.979i)25-s + 1.92i·29-s − 1.79·31-s + (−1.43 − 1.17i)35-s − 2.42·49-s + 1.94·53-s + (−0.809 − 0.660i)55-s + 1.35i·59-s − 1.14i·73-s − 1.93·77-s + 1.12·79-s + 0.620·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.632 + 0.774i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $-0.632 + 0.774i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (1009, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ -0.632 + 0.774i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.593442329\)
\(L(\frac12)\) \(\approx\) \(1.593442329\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.41 + 1.73i)T \)
good7 \( 1 + 4.89iT - 7T^{2} \)
11 \( 1 + 3.46iT - 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 - 23T^{2} \)
29 \( 1 - 10.3iT - 29T^{2} \)
31 \( 1 + 10T + 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 + 41T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 14.1T + 53T^{2} \)
59 \( 1 - 10.3iT - 59T^{2} \)
61 \( 1 - 61T^{2} \)
67 \( 1 + 67T^{2} \)
71 \( 1 + 71T^{2} \)
73 \( 1 + 9.79iT - 73T^{2} \)
79 \( 1 - 10T + 79T^{2} \)
83 \( 1 - 5.65T + 83T^{2} \)
89 \( 1 + 89T^{2} \)
97 \( 1 + 19.5iT - 97T^{2} \)
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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

−9.149391909632970242267429015395, −8.580115013620057442193123971387, −7.53787740900187822290364337611, −6.91722019354896981780305752541, −5.87174692736383901084906681931, −5.05922089051886962017136704070, −4.08812953444809035284952661395, −3.28943928624846235276789453245, −1.62128297620107574046629900470, −0.63047446408072178130645993478, 2.06472893292625588889635423436, 2.41776521036936597091717866246, 3.70013735610571574852967408864, 5.05650150553598078445832468228, 5.72972707891141567591415584032, 6.43588585262053776106561951387, 7.34266705582853022693734802159, 8.266992419445714041228980262831, 9.280337106619467449776902529735, 9.590378136167846634026406152627

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