Properties

Label 2-1536-12.11-c1-0-12
Degree $2$
Conductor $1536$
Sign $-0.962 + 0.270i$
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  + (0.468 + 1.66i)3-s + 3.02i·5-s + 3.62i·7-s + (−2.56 + 1.56i)9-s − 3.33·11-s + 1.32·13-s + (−5.03 + 1.41i)15-s + 7.12i·17-s − 5.20i·19-s + (−6.04 + 1.69i)21-s + 4.41·23-s − 4.12·25-s + (−3.80 − 3.54i)27-s − 6.41i·29-s − 0.794i·31-s + ⋯
L(s)  = 1  + (0.270 + 0.962i)3-s + 1.35i·5-s + 1.36i·7-s + (−0.853 + 0.520i)9-s − 1.00·11-s + 0.367·13-s + (−1.30 + 0.365i)15-s + 1.72i·17-s − 1.19i·19-s + (−1.31 + 0.370i)21-s + 0.920·23-s − 0.824·25-s + (−0.731 − 0.681i)27-s − 1.19i·29-s − 0.142i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.962 + 0.270i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.962 + 0.270i$
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: \((2,\ 1536,\ (\ :1/2),\ -0.962 + 0.270i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.455902194\)
\(L(\frac12)\) \(\approx\) \(1.455902194\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.468 - 1.66i)T \)
good5 \( 1 - 3.02iT - 5T^{2} \)
7 \( 1 - 3.62iT - 7T^{2} \)
11 \( 1 + 3.33T + 11T^{2} \)
13 \( 1 - 1.32T + 13T^{2} \)
17 \( 1 - 7.12iT - 17T^{2} \)
19 \( 1 + 5.20iT - 19T^{2} \)
23 \( 1 - 4.41T + 23T^{2} \)
29 \( 1 + 6.41iT - 29T^{2} \)
31 \( 1 + 0.794iT - 31T^{2} \)
37 \( 1 - 8.10T + 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 1.46iT - 43T^{2} \)
47 \( 1 - 5.65T + 47T^{2} \)
53 \( 1 - 3.76iT - 53T^{2} \)
59 \( 1 + 5.73T + 59T^{2} \)
61 \( 1 - 13.4T + 61T^{2} \)
67 \( 1 + 7.08iT - 67T^{2} \)
71 \( 1 + 10.0T + 71T^{2} \)
73 \( 1 - 10.2T + 73T^{2} \)
79 \( 1 + 4.86iT - 79T^{2} \)
83 \( 1 - 0.410T + 83T^{2} \)
89 \( 1 - 4.87iT - 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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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

−9.895231075836523388772744348235, −9.153723170078416800854508963086, −8.372133439972329329036624365430, −7.69072255856444150754946037787, −6.40582315407733859175121327844, −5.85936271537461284115617605669, −4.93748189477817703246920600889, −3.83615829475051786029624806930, −2.75906390340483858348717149787, −2.43807067415080929251834228558, 0.58103646907447817740982229299, 1.32430756383840911539612711523, 2.76293076595523426218008613761, 3.88988095140659182818568480894, 4.96082324541542417153836949813, 5.60010257080055007540951460674, 6.89206896871466701826782052632, 7.45065913655568624134061529751, 8.139878979052528794363501186141, 8.874192214310794368049010221099

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