Properties

Label 2-3072-1.1-c1-0-12
Degree $2$
Conductor $3072$
Sign $1$
Analytic cond. $24.5300$
Root an. cond. $4.95278$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.79·5-s + 0.158·7-s + 9-s + 5.37·11-s + 5.95·13-s + 1.79·15-s − 3.05·17-s − 3.05·19-s − 0.158·21-s + 2.82·23-s − 1.76·25-s − 27-s + 2.96·29-s − 4.15·31-s − 5.37·33-s − 0.285·35-s + 8.46·37-s − 5.95·39-s + 2.60·41-s − 8.13·43-s − 1.79·45-s − 2.82·47-s − 6.97·49-s + 3.05·51-s + 5.03·53-s − 9.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.804·5-s + 0.0600·7-s + 0.333·9-s + 1.61·11-s + 1.65·13-s + 0.464·15-s − 0.740·17-s − 0.700·19-s − 0.0346·21-s + 0.589·23-s − 0.353·25-s − 0.192·27-s + 0.551·29-s − 0.746·31-s − 0.934·33-s − 0.0483·35-s + 1.39·37-s − 0.953·39-s + 0.406·41-s − 1.24·43-s − 0.268·45-s − 0.412·47-s − 0.996·49-s + 0.427·51-s + 0.690·53-s − 1.30·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $1$
Analytic conductor: \(24.5300\)
Root analytic conductor: \(4.95278\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.445038188\)
\(L(\frac12)\) \(\approx\) \(1.445038188\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
good5 \( 1 + 1.79T + 5T^{2} \)
7 \( 1 - 0.158T + 7T^{2} \)
11 \( 1 - 5.37T + 11T^{2} \)
13 \( 1 - 5.95T + 13T^{2} \)
17 \( 1 + 3.05T + 17T^{2} \)
19 \( 1 + 3.05T + 19T^{2} \)
23 \( 1 - 2.82T + 23T^{2} \)
29 \( 1 - 2.96T + 29T^{2} \)
31 \( 1 + 4.15T + 31T^{2} \)
37 \( 1 - 8.46T + 37T^{2} \)
41 \( 1 - 2.60T + 41T^{2} \)
43 \( 1 + 8.13T + 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 - 5.03T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 - 5.18T + 61T^{2} \)
67 \( 1 + 1.08T + 67T^{2} \)
71 \( 1 + 0.317T + 71T^{2} \)
73 \( 1 - 1.33T + 73T^{2} \)
79 \( 1 + 9.69T + 79T^{2} \)
83 \( 1 - 0.163T + 83T^{2} \)
89 \( 1 - 14.3T + 89T^{2} \)
97 \( 1 + 0.571T + 97T^{2} \)
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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

−8.690587493356687688892576154933, −8.064139847851923828163048632952, −6.99860063250874918767491375094, −6.45914446314031930501491690476, −5.87210147855211633331650705514, −4.63598009374082431147807944454, −4.02891165608713847351015565897, −3.40283064774786394980810778129, −1.80983909119701770521804643619, −0.78780356777455354705024187437, 0.78780356777455354705024187437, 1.80983909119701770521804643619, 3.40283064774786394980810778129, 4.02891165608713847351015565897, 4.63598009374082431147807944454, 5.87210147855211633331650705514, 6.45914446314031930501491690476, 6.99860063250874918767491375094, 8.064139847851923828163048632952, 8.690587493356687688892576154933

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