Properties

Label 2-3072-1.1-c1-0-34
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.79·5-s + 2.15·7-s + 9-s − 2.54·11-s + 1.95·13-s + 3.79·15-s + 0.224·17-s + 0.224·19-s − 2.15·21-s − 2.82·23-s + 9.42·25-s − 27-s + 2.62·29-s + 1.84·31-s + 2.54·33-s − 8.19·35-s − 5.18·37-s − 1.95·39-s + 5.88·41-s + 10.9·43-s − 3.79·45-s + 2.82·47-s − 2.33·49-s − 0.224·51-s − 10.6·53-s + 9.65·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.69·5-s + 0.816·7-s + 0.333·9-s − 0.766·11-s + 0.542·13-s + 0.980·15-s + 0.0545·17-s + 0.0515·19-s − 0.471·21-s − 0.589·23-s + 1.88·25-s − 0.192·27-s + 0.487·29-s + 0.330·31-s + 0.442·33-s − 1.38·35-s − 0.853·37-s − 0.313·39-s + 0.918·41-s + 1.67·43-s − 0.566·45-s + 0.412·47-s − 0.334·49-s − 0.0314·51-s − 1.45·53-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: \(1\)
Selberg data: \((2,\ 3072,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.79T + 5T^{2} \)
7 \( 1 - 2.15T + 7T^{2} \)
11 \( 1 + 2.54T + 11T^{2} \)
13 \( 1 - 1.95T + 13T^{2} \)
17 \( 1 - 0.224T + 17T^{2} \)
19 \( 1 - 0.224T + 19T^{2} \)
23 \( 1 + 2.82T + 23T^{2} \)
29 \( 1 - 2.62T + 29T^{2} \)
31 \( 1 - 1.84T + 31T^{2} \)
37 \( 1 + 5.18T + 37T^{2} \)
41 \( 1 - 5.88T + 41T^{2} \)
43 \( 1 - 10.9T + 43T^{2} \)
47 \( 1 - 2.82T + 47T^{2} \)
53 \( 1 + 10.6T + 53T^{2} \)
59 \( 1 - 5.65T + 59T^{2} \)
61 \( 1 + 8.46T + 61T^{2} \)
67 \( 1 - 14.7T + 67T^{2} \)
71 \( 1 + 4.31T + 71T^{2} \)
73 \( 1 - 5.97T + 73T^{2} \)
79 \( 1 + 15.0T + 79T^{2} \)
83 \( 1 + 14.3T + 83T^{2} \)
89 \( 1 + 1.42T + 89T^{2} \)
97 \( 1 + 16.3T + 97T^{2} \)
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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

−8.127051134195957356095330421642, −7.71504836790309885652979779485, −7.00490939200502567191487726359, −6.00875653261882002845927317322, −5.11519260331973458721246701333, −4.40567691563585988635624126163, −3.78337667086529487549485514100, −2.68261032884944676707285555881, −1.20492633795042585721050146197, 0, 1.20492633795042585721050146197, 2.68261032884944676707285555881, 3.78337667086529487549485514100, 4.40567691563585988635624126163, 5.11519260331973458721246701333, 6.00875653261882002845927317322, 7.00490939200502567191487726359, 7.71504836790309885652979779485, 8.127051134195957356095330421642

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