Properties

Label 2-3072-3.2-c0-0-1
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

\(L(\frac{1}{2})\) \(\approx\) \(1.163767079\)
\(L(\frac12)\) \(\approx\) \(1.163767079\)
\(L(1)\) 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

−8.699632468882944135308092720186, −8.238381968969571126504455675225, −7.24435152408793876969587731719, −6.66710307341256271006750295899, −5.63281229772503130206451191561, −5.21914527619629865995098436437, −4.33659027224392614150832295957, −3.54574700704200518012477746876, −1.94112377061757138325208870068, −1.13500180564826304102144554184, 1.13500180564826304102144554184, 1.94112377061757138325208870068, 3.54574700704200518012477746876, 4.33659027224392614150832295957, 5.21914527619629865995098436437, 5.63281229772503130206451191561, 6.66710307341256271006750295899, 7.24435152408793876969587731719, 8.238381968969571126504455675225, 8.699632468882944135308092720186

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