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Properties

Label 2-1440-72.11-c1-0-23
Degree $2$
Conductor $1440$
Sign $0.722 - 0.691i$
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  + (0.0300 + 1.73i)3-s + (−0.5 − 0.866i)5-s + (3.45 + 1.99i)7-s + (−2.99 + 0.104i)9-s + (−0.520 − 0.300i)11-s + (4.44 − 2.56i)13-s + (1.48 − 0.891i)15-s − 5.37i·17-s + 8.29·19-s + (−3.34 + 6.03i)21-s + (0.0667 + 0.115i)23-s + (−0.499 + 0.866i)25-s + (−0.270 − 5.18i)27-s + (−0.507 + 0.879i)29-s + (−1.58 + 0.914i)31-s + ⋯
L(s)  = 1  + (0.0173 + 0.999i)3-s + (−0.223 − 0.387i)5-s + (1.30 + 0.753i)7-s + (−0.999 + 0.0347i)9-s + (−0.157 − 0.0906i)11-s + (1.23 − 0.711i)13-s + (0.383 − 0.230i)15-s − 1.30i·17-s + 1.90·19-s + (−0.730 + 1.31i)21-s + (0.0139 + 0.0241i)23-s + (−0.0999 + 0.173i)25-s + (−0.0520 − 0.998i)27-s + (−0.0942 + 0.163i)29-s + (−0.284 + 0.164i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.722 - 0.691i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.722 - 0.691i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (911, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.722 - 0.691i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.992334665\)
\(L(\frac12)\) \(\approx\) \(1.992334665\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.0300 - 1.73i)T \)
5 \( 1 + (0.5 + 0.866i)T \)
good7 \( 1 + (-3.45 - 1.99i)T + (3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.520 + 0.300i)T + (5.5 + 9.52i)T^{2} \)
13 \( 1 + (-4.44 + 2.56i)T + (6.5 - 11.2i)T^{2} \)
17 \( 1 + 5.37iT - 17T^{2} \)
19 \( 1 - 8.29T + 19T^{2} \)
23 \( 1 + (-0.0667 - 0.115i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.507 - 0.879i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.58 - 0.914i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 - 5.87iT - 37T^{2} \)
41 \( 1 + (-4.43 + 2.56i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.36 + 5.82i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.27 - 9.13i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 - 6.25T + 53T^{2} \)
59 \( 1 + (10.1 - 5.87i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + (2.68 + 1.55i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (4.14 + 7.17i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 - 10.6T + 71T^{2} \)
73 \( 1 + 6.71T + 73T^{2} \)
79 \( 1 + (-7.14 - 4.12i)T + (39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.85 - 1.65i)T + (41.5 + 71.8i)T^{2} \)
89 \( 1 - 3.54iT - 89T^{2} \)
97 \( 1 + (-2.33 + 4.04i)T + (-48.5 - 84.0i)T^{2} \)
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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

−9.395636405313592058651088347557, −8.969970230897999383216371643282, −8.111862919837765189610764182382, −7.55305935761431573566310808596, −5.97162664383143414745339711500, −5.23513940762764682094950584673, −4.82654142122105898196019290896, −3.60514963250307084319038444154, −2.73165466200079303496490828151, −1.11589242455334138924969732973, 1.11289415510441327881434650001, 1.89331538073181113322502816972, 3.34739170125842198784495154179, 4.23922703472414375177768592806, 5.43553646257011319425135596183, 6.24739718552107550333916263187, 7.18886454262809770570680562497, 7.77346699928697403975215549095, 8.336911086459054816860835034841, 9.257502652486309941528335568618

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