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Properties

Label 2-1440-5.4-c3-0-59
Degree $2$
Conductor $1440$
Sign $0.640 + 0.767i$
Analytic cond. $84.9627$
Root an. cond. $9.21752$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−8.58 + 7.16i)5-s − 28.3i·7-s + 65.2·11-s − 33.6i·13-s + 73.3i·17-s + 134.·19-s + 14.7i·23-s + (22.3 − 122. i)25-s − 224.·29-s − 68.8·31-s + (202. + 243. i)35-s + 196. i·37-s + 143.·41-s + 15.0i·43-s + 134. i·47-s + ⋯
L(s)  = 1  + (−0.767 + 0.640i)5-s − 1.52i·7-s + 1.78·11-s − 0.718i·13-s + 1.04i·17-s + 1.61·19-s + 0.133i·23-s + (0.178 − 0.983i)25-s − 1.43·29-s − 0.398·31-s + (0.979 + 1.17i)35-s + 0.872i·37-s + 0.545·41-s + 0.0534i·43-s + 0.417i·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.640 + 0.767i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.640 + 0.767i$
Analytic conductor: \(84.9627\)
Root analytic conductor: \(9.21752\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (289, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :3/2),\ 0.640 + 0.767i)\)

Particular Values

\(L(2)\) \(\approx\) \(1.998296291\)
\(L(\frac12)\) \(\approx\) \(1.998296291\)
\(L(\frac{5}{2})\) 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 \( 1 + (8.58 - 7.16i)T \)
good7 \( 1 + 28.3iT - 343T^{2} \)
11 \( 1 - 65.2T + 1.33e3T^{2} \)
13 \( 1 + 33.6iT - 2.19e3T^{2} \)
17 \( 1 - 73.3iT - 4.91e3T^{2} \)
19 \( 1 - 134.T + 6.85e3T^{2} \)
23 \( 1 - 14.7iT - 1.21e4T^{2} \)
29 \( 1 + 224.T + 2.43e4T^{2} \)
31 \( 1 + 68.8T + 2.97e4T^{2} \)
37 \( 1 - 196. iT - 5.06e4T^{2} \)
41 \( 1 - 143.T + 6.89e4T^{2} \)
43 \( 1 - 15.0iT - 7.95e4T^{2} \)
47 \( 1 - 134. iT - 1.03e5T^{2} \)
53 \( 1 + 262. iT - 1.48e5T^{2} \)
59 \( 1 - 119.T + 2.05e5T^{2} \)
61 \( 1 - 16.5T + 2.26e5T^{2} \)
67 \( 1 + 545. iT - 3.00e5T^{2} \)
71 \( 1 + 199.T + 3.57e5T^{2} \)
73 \( 1 - 43.2iT - 3.89e5T^{2} \)
79 \( 1 - 438.T + 4.93e5T^{2} \)
83 \( 1 + 1.22e3iT - 5.71e5T^{2} \)
89 \( 1 - 723.T + 7.04e5T^{2} \)
97 \( 1 - 1.13e3iT - 9.12e5T^{2} \)
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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

−9.128831560725129775758468624750, −7.974532122857053507935299634760, −7.43732265132176471546070038974, −6.78622486322649681410690078639, −5.93459929366125543139677946006, −4.58132878198676697965991392999, −3.65984965634900910398055854886, −3.42371333599497566894840840404, −1.53040152235286694191464169871, −0.60056120893212819174337816022, 0.916420813942254349472207768368, 2.02102699726267411937159830675, 3.31075046706515923633951982129, 4.14798981091642938559465305181, 5.17369215844217033729553371373, 5.83859592585846246485813520324, 6.94832739312943683883370088778, 7.62479217509087790643784072354, 8.795507949261063672295523240467, 9.189187629661858420910846663069

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