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Properties

Label 2-1536-12.11-c1-0-24
Degree $2$
Conductor $1536$
Sign $0.912 + 0.408i$
Analytic cond. $12.2650$
Root an. cond. $3.50214$
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  + (−1.58 − 0.707i)3-s − 2.82i·5-s + (2.00 + 2.23i)9-s + 3.16·11-s + 6.32·13-s + (−2.00 + 4.47i)15-s + 4.47i·17-s + 4.24i·19-s − 4·23-s − 3.00·25-s + (−1.58 − 4.94i)27-s + 2.82i·29-s + 8.94i·31-s + (−5.00 − 2.23i)33-s + 6.32·37-s + ⋯
L(s)  = 1  + (−0.912 − 0.408i)3-s − 1.26i·5-s + (0.666 + 0.745i)9-s + 0.953·11-s + 1.75·13-s + (−0.516 + 1.15i)15-s + 1.08i·17-s + 0.973i·19-s − 0.834·23-s − 0.600·25-s + (−0.304 − 0.952i)27-s + 0.525i·29-s + 1.60i·31-s + (−0.870 − 0.389i)33-s + 1.03·37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.912 + 0.408i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $0.912 + 0.408i$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (1535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ 0.912 + 0.408i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.426966317\)
\(L(\frac12)\) \(\approx\) \(1.426966317\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.58 + 0.707i)T \)
good5 \( 1 + 2.82iT - 5T^{2} \)
7 \( 1 - 7T^{2} \)
11 \( 1 - 3.16T + 11T^{2} \)
13 \( 1 - 6.32T + 13T^{2} \)
17 \( 1 - 4.47iT - 17T^{2} \)
19 \( 1 - 4.24iT - 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 2.82iT - 29T^{2} \)
31 \( 1 - 8.94iT - 31T^{2} \)
37 \( 1 - 6.32T + 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 - 7.07iT - 43T^{2} \)
47 \( 1 - 8T + 47T^{2} \)
53 \( 1 + 8.48iT - 53T^{2} \)
59 \( 1 + 9.48T + 59T^{2} \)
61 \( 1 - 6.32T + 61T^{2} \)
67 \( 1 + 4.24iT - 67T^{2} \)
71 \( 1 - 12T + 71T^{2} \)
73 \( 1 + 4T + 73T^{2} \)
79 \( 1 - 8.94iT - 79T^{2} \)
83 \( 1 - 3.16T + 83T^{2} \)
89 \( 1 + 4.47iT - 89T^{2} \)
97 \( 1 + 12T + 97T^{2} \)
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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

−9.298004473226386619263703654570, −8.475775338852020593365865247881, −8.032782675253354913666955173163, −6.72175253642393160329086050210, −6.07026991624816301116196546047, −5.46296157522766276366388933113, −4.35217973711021216271720565551, −3.73147980194962270142867006084, −1.61017655821101794869878385782, −1.10838271084235557111784262997, 0.846266450321152904558089093536, 2.51475974876372517779612561254, 3.72708384324553953696022622201, 4.27276073337075573304067774885, 5.65942182250206169698231072932, 6.25960204029362143842404245042, 6.85153866718181281366528285608, 7.69016025618744050708697124652, 8.994436231681942898359152564187, 9.542709625237692157925301210593

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