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Properties

Label 2-1440-9.4-c1-0-4
Degree $2$
Conductor $1440$
Sign $0.381 - 0.924i$
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  + (−1.66 − 0.483i)3-s + (0.5 − 0.866i)5-s + (−1.56 − 2.70i)7-s + (2.53 + 1.60i)9-s + (−0.412 − 0.714i)11-s + (−3.03 + 5.25i)13-s + (−1.25 + 1.19i)15-s − 2.74·17-s − 0.824·19-s + (1.28 + 5.25i)21-s + (−1.76 + 3.05i)23-s + (−0.499 − 0.866i)25-s + (−3.43 − 3.90i)27-s + (3.15 + 5.47i)29-s + (3.63 − 6.29i)31-s + ⋯
L(s)  = 1  + (−0.960 − 0.279i)3-s + (0.223 − 0.387i)5-s + (−0.589 − 1.02i)7-s + (0.843 + 0.536i)9-s + (−0.124 − 0.215i)11-s + (−0.840 + 1.45i)13-s + (−0.322 + 0.309i)15-s − 0.665·17-s − 0.189·19-s + (0.280 + 1.14i)21-s + (−0.368 + 0.637i)23-s + (−0.0999 − 0.173i)25-s + (−0.660 − 0.750i)27-s + (0.586 + 1.01i)29-s + (0.653 − 1.13i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1440 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.381 - 0.924i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1440\)    =    \(2^{5} \cdot 3^{2} \cdot 5\)
Sign: $0.381 - 0.924i$
Analytic conductor: \(11.4984\)
Root analytic conductor: \(3.39093\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1440} (481, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1440,\ (\ :1/2),\ 0.381 - 0.924i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5984851823\)
\(L(\frac12)\) \(\approx\) \(0.5984851823\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.66 + 0.483i)T \)
5 \( 1 + (-0.5 + 0.866i)T \)
good7 \( 1 + (1.56 + 2.70i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (0.412 + 0.714i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (3.03 - 5.25i)T + (-6.5 - 11.2i)T^{2} \)
17 \( 1 + 2.74T + 17T^{2} \)
19 \( 1 + 0.824T + 19T^{2} \)
23 \( 1 + (1.76 - 3.05i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.15 - 5.47i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.63 + 6.29i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + 2.06T + 37T^{2} \)
41 \( 1 + (-2.78 + 4.82i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-3.73 - 6.47i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.75 - 6.49i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + 1.48T + 53T^{2} \)
59 \( 1 + (-0.722 + 1.25i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-4.19 - 7.25i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (5.91 - 10.2i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 13.4T + 73T^{2} \)
79 \( 1 + (-7.06 - 12.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2.18 - 3.77i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 2.83T + 89T^{2} \)
97 \( 1 + (-2.65 - 4.60i)T + (-48.5 + 84.0i)T^{2} \)
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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

−9.738937292830537296424106126984, −9.096118149703571130120187422984, −7.84965122707360968178064988291, −7.03716348820812637539255368905, −6.54151591941583317507542240268, −5.61549183529496195893160793204, −4.55793838038445800208377448812, −4.04413666508983015008919571942, −2.35869872522381109035034876048, −1.08579834130707174400969488039, 0.30997653231755930813603077536, 2.26343233826018841211157443156, 3.16690117226696697482243714584, 4.51374501815373260778679476146, 5.32064295619621861243921792478, 6.06687241314826532013952231429, 6.67902418738641153812507017433, 7.67328240108572720165931017567, 8.695820930314016358817581336821, 9.587287408774316948330173144936

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