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Properties

Label 2-1536-12.11-c1-0-11
Degree $2$
Conductor $1536$
Sign $-0.489 - 0.871i$
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  + (−1.51 + 0.848i)3-s + 0.936i·5-s + 2.20i·7-s + (1.56 − 2.56i)9-s − 1.69·11-s + 4.27·13-s + (−0.794 − 1.41i)15-s − 1.12i·17-s + 4.34i·19-s + (−1.87 − 3.33i)21-s + 7.24·23-s + 4.12·25-s + (−0.185 + 5.19i)27-s + 5.73i·29-s − 5.03i·31-s + ⋯
L(s)  = 1  + (−0.871 + 0.489i)3-s + 0.418i·5-s + 0.834i·7-s + (0.520 − 0.853i)9-s − 0.511·11-s + 1.18·13-s + (−0.205 − 0.365i)15-s − 0.272i·17-s + 0.996i·19-s + (−0.408 − 0.727i)21-s + 1.51·23-s + 0.824·25-s + (−0.0357 + 0.999i)27-s + 1.06i·29-s − 0.904i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1536 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.489 - 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $-0.489 - 0.871i$
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: \((2,\ 1536,\ (\ :1/2),\ -0.489 - 0.871i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.105550043\)
\(L(\frac12)\) \(\approx\) \(1.105550043\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 - 0.848i)T \)
good5 \( 1 - 0.936iT - 5T^{2} \)
7 \( 1 - 2.20iT - 7T^{2} \)
11 \( 1 + 1.69T + 11T^{2} \)
13 \( 1 - 4.27T + 13T^{2} \)
17 \( 1 + 1.12iT - 17T^{2} \)
19 \( 1 - 4.34iT - 19T^{2} \)
23 \( 1 - 7.24T + 23T^{2} \)
29 \( 1 - 5.73iT - 29T^{2} \)
31 \( 1 + 5.03iT - 31T^{2} \)
37 \( 1 + 9.06T + 37T^{2} \)
41 \( 1 - 4iT - 41T^{2} \)
43 \( 1 - 7.73iT - 43T^{2} \)
47 \( 1 + 5.65T + 47T^{2} \)
53 \( 1 + 14.2iT - 53T^{2} \)
59 \( 1 + 6.41T + 59T^{2} \)
61 \( 1 - 8.01T + 61T^{2} \)
67 \( 1 - 10.3iT - 67T^{2} \)
71 \( 1 + 1.58T + 71T^{2} \)
73 \( 1 + 6.24T + 73T^{2} \)
79 \( 1 - 10.6iT - 79T^{2} \)
83 \( 1 + 13.7T + 83T^{2} \)
89 \( 1 - 13.1iT - 89T^{2} \)
97 \( 1 + 7.12T + 97T^{2} \)
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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

−9.838305838378018165031518973999, −8.952553858707618119925880586241, −8.296741012424154228327925853702, −7.05224861095398788089106530668, −6.41020418882487983145663618455, −5.52616262137251734802508001715, −4.98696915658842632728518446641, −3.73637159629481647722640520934, −2.90171279746219767716314591213, −1.31590785820848960430885200383, 0.55749212819604750355005658960, 1.55055608940724339007138750920, 3.09799192634161882433610103784, 4.34151128329251640839142645256, 5.06198151663038431190428520592, 5.90711716796236350633233106149, 6.90124820882345225840033526287, 7.30430421441166015147922792813, 8.433347665605143532993476897924, 9.022340986551733185236210224444

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