Properties

Label 2-1536-8.5-c1-0-13
Degree $2$
Conductor $1536$
Sign $1$
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  + i·3-s − 0.585i·5-s − 3.41·7-s − 9-s − 2i·11-s + 2.82i·13-s + 0.585·15-s + 3.65·17-s − 5.65i·19-s − 3.41i·21-s − 1.17·23-s + 4.65·25-s − i·27-s + 0.585i·29-s + 4.58·31-s + ⋯
L(s)  = 1  + 0.577i·3-s − 0.261i·5-s − 1.29·7-s − 0.333·9-s − 0.603i·11-s + 0.784i·13-s + 0.151·15-s + 0.886·17-s − 1.29i·19-s − 0.745i·21-s − 0.244·23-s + 0.931·25-s − 0.192i·27-s + 0.108i·29-s + 0.823·31-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(1536\)    =    \(2^{9} \cdot 3\)
Sign: $1$
Analytic conductor: \(12.2650\)
Root analytic conductor: \(3.50214\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1536} (769, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1536,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.376685892\)
\(L(\frac12)\) \(\approx\) \(1.376685892\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
good5 \( 1 + 0.585iT - 5T^{2} \)
7 \( 1 + 3.41T + 7T^{2} \)
11 \( 1 + 2iT - 11T^{2} \)
13 \( 1 - 2.82iT - 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 5.65iT - 19T^{2} \)
23 \( 1 + 1.17T + 23T^{2} \)
29 \( 1 - 0.585iT - 29T^{2} \)
31 \( 1 - 4.58T + 31T^{2} \)
37 \( 1 - 9.65iT - 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 + 1.65iT - 43T^{2} \)
47 \( 1 - 12.4T + 47T^{2} \)
53 \( 1 + 11.8iT - 53T^{2} \)
59 \( 1 - 4iT - 59T^{2} \)
61 \( 1 + 9.65iT - 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 9.17T + 71T^{2} \)
73 \( 1 - 1.65T + 73T^{2} \)
79 \( 1 - 5.75T + 79T^{2} \)
83 \( 1 - 9.31iT - 83T^{2} \)
89 \( 1 + 2T + 89T^{2} \)
97 \( 1 - 13.3T + 97T^{2} \)
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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

−9.362343567005639172030083570546, −8.951955517105535879197790904475, −7.971820089118118289789832604895, −6.83579435721434827338561810681, −6.27109582918201779910648328658, −5.28032299262245179452380244487, −4.38208042267519582795814839890, −3.39775947061958694993848936619, −2.64097456115092330237852405054, −0.74591441958885828770525285792, 0.919227192973046285345743900194, 2.46685874797301193814356934110, 3.26220628638032236354162177701, 4.26759341834487760253298501176, 5.81349288483746806590914092557, 5.99147508917696044031915295733, 7.24565716450861065065094149492, 7.56926843753550460606630417946, 8.654921430170131458752864256980, 9.534766737856858134885186883381

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