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Properties

Label 2-3072-96.53-c0-0-5
Degree $2$
Conductor $3072$
Sign $0.195 + 0.980i$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.541 − 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (0.541 + 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + 1.84·31-s + (0.292 − 0.707i)37-s + (−1.30 − 1.30i)39-s + (−1.30 − 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 0.707i)61-s + ⋯
L(s)  = 1  + (0.923 − 0.382i)3-s + (−0.541 − 0.541i)7-s + (0.707 − 0.707i)9-s + (−0.707 − 1.70i)13-s + (0.541 + 1.30i)19-s + (−0.707 − 0.292i)21-s + (−0.707 − 0.707i)25-s + (0.382 − 0.923i)27-s + 1.84·31-s + (0.292 − 0.707i)37-s + (−1.30 − 1.30i)39-s + (−1.30 − 0.541i)43-s − 0.414i·49-s + (1 + 0.999i)57-s + (−1.70 + 0.707i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3072 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.195 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $0.195 + 0.980i$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1409, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 0.195 + 0.980i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.544577906\)
\(L(\frac12)\) \(\approx\) \(1.544577906\)
\(L(1)\) 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 \( 1 + (-0.923 + 0.382i)T \)
good5 \( 1 + (0.707 + 0.707i)T^{2} \)
7 \( 1 + (0.541 + 0.541i)T + iT^{2} \)
11 \( 1 + (-0.707 - 0.707i)T^{2} \)
13 \( 1 + (0.707 + 1.70i)T + (-0.707 + 0.707i)T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 + (-0.541 - 1.30i)T + (-0.707 + 0.707i)T^{2} \)
23 \( 1 + iT^{2} \)
29 \( 1 + (-0.707 + 0.707i)T^{2} \)
31 \( 1 - 1.84T + T^{2} \)
37 \( 1 + (-0.292 + 0.707i)T + (-0.707 - 0.707i)T^{2} \)
41 \( 1 + iT^{2} \)
43 \( 1 + (1.30 + 0.541i)T + (0.707 + 0.707i)T^{2} \)
47 \( 1 + T^{2} \)
53 \( 1 + (-0.707 - 0.707i)T^{2} \)
59 \( 1 + (0.707 + 0.707i)T^{2} \)
61 \( 1 + (1.70 - 0.707i)T + (0.707 - 0.707i)T^{2} \)
67 \( 1 + (0.707 - 0.707i)T^{2} \)
71 \( 1 - iT^{2} \)
73 \( 1 + (-1 + i)T - iT^{2} \)
79 \( 1 - 1.84iT - T^{2} \)
83 \( 1 + (0.707 - 0.707i)T^{2} \)
89 \( 1 - iT^{2} \)
97 \( 1 - 1.41T + T^{2} \)
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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

−8.517477076752881233648046376358, −7.898887645544288949634113390970, −7.50557135132295737706925429039, −6.54058985594191227446435093493, −5.82823757565107277488013114432, −4.76685915999708901069319915785, −3.71568569842177434943006997680, −3.13849558269152225352605901176, −2.21809841145705064541057002240, −0.858937579820070035753671012287, 1.72020134570779160292957692608, 2.65802171947019341840769462443, 3.32460275381554217259945311386, 4.51744464845056833147256880783, 4.85004853368689502548182506772, 6.18108515530931897751560160368, 6.87779168814805950881626545380, 7.58747178247757437681551562439, 8.464463562305884257064589513413, 9.190838925317730060968527399837

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