Properties

Label 2-3072-3.2-c0-0-3
Degree $2$
Conductor $3072$
Sign $1$
Analytic cond. $1.53312$
Root an. cond. $1.23819$
Motivic weight $0$
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 + 1.41·7-s + 9-s − 1.41·13-s + 1.41·21-s + 25-s + 27-s − 1.41·31-s + 1.41·37-s − 1.41·39-s + 1.00·49-s − 1.41·61-s + 1.41·63-s − 2·67-s + 75-s − 1.41·79-s + 81-s − 2.00·91-s − 1.41·93-s − 1.41·103-s + 1.41·109-s + 1.41·111-s − 1.41·117-s + ⋯
L(s)  = 1  + 3-s + 1.41·7-s + 9-s − 1.41·13-s + 1.41·21-s + 25-s + 27-s − 1.41·31-s + 1.41·37-s − 1.41·39-s + 1.00·49-s − 1.41·61-s + 1.41·63-s − 2·67-s + 75-s − 1.41·79-s + 81-s − 2.00·91-s − 1.41·93-s − 1.41·103-s + 1.41·109-s + 1.41·111-s − 1.41·117-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(3072\)    =    \(2^{10} \cdot 3\)
Sign: $1$
Analytic conductor: \(1.53312\)
Root analytic conductor: \(1.23819\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{3072} (1025, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 3072,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.024070258\)
\(L(\frac12)\) \(\approx\) \(2.024070258\)
\(L(1)\) 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 \( 1 - T \)
good5 \( 1 - T^{2} \)
7 \( 1 - 1.41T + T^{2} \)
11 \( 1 - T^{2} \)
13 \( 1 + 1.41T + T^{2} \)
17 \( 1 - T^{2} \)
19 \( 1 + T^{2} \)
23 \( 1 - T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + 1.41T + T^{2} \)
37 \( 1 - 1.41T + T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - T^{2} \)
59 \( 1 - T^{2} \)
61 \( 1 + 1.41T + T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + T^{2} \)
79 \( 1 + 1.41T + T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 - T^{2} \)
97 \( 1 + T^{2} \)
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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

−8.881834692795028887061049922517, −8.072541743793536733190002331900, −7.52596335611397931786514473452, −7.01086211349315982832412088913, −5.72619460153615484409381773746, −4.74555622103127010106487657766, −4.39519548484192209994284929324, −3.14120914875801111181149470575, −2.30665643281194984544791808894, −1.44212126373452317595691176032, 1.44212126373452317595691176032, 2.30665643281194984544791808894, 3.14120914875801111181149470575, 4.39519548484192209994284929324, 4.74555622103127010106487657766, 5.72619460153615484409381773746, 7.01086211349315982832412088913, 7.52596335611397931786514473452, 8.072541743793536733190002331900, 8.881834692795028887061049922517

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