Properties

Label 2-8046-1.1-c1-0-172
Degree $2$
Conductor $8046$
Sign $-1$
Analytic cond. $64.2476$
Root an. cond. $8.01546$
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  + 2-s + 4-s + 1.12·5-s − 0.686·7-s + 8-s + 1.12·10-s − 3.05·11-s − 3.99·13-s − 0.686·14-s + 16-s − 5.80·17-s + 8.32·19-s + 1.12·20-s − 3.05·22-s + 2.20·23-s − 3.73·25-s − 3.99·26-s − 0.686·28-s − 6.02·29-s + 8.16·31-s + 32-s − 5.80·34-s − 0.772·35-s + 4.59·37-s + 8.32·38-s + 1.12·40-s − 5.82·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 0.503·5-s − 0.259·7-s + 0.353·8-s + 0.355·10-s − 0.920·11-s − 1.10·13-s − 0.183·14-s + 0.250·16-s − 1.40·17-s + 1.90·19-s + 0.251·20-s − 0.651·22-s + 0.459·23-s − 0.746·25-s − 0.782·26-s − 0.129·28-s − 1.11·29-s + 1.46·31-s + 0.176·32-s − 0.994·34-s − 0.130·35-s + 0.754·37-s + 1.34·38-s + 0.177·40-s − 0.910·41-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8046 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8046\)    =    \(2 \cdot 3^{3} \cdot 149\)
Sign: $-1$
Analytic conductor: \(64.2476\)
Root analytic conductor: \(8.01546\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ -1)\)

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 \( 1 - T \)
3 \( 1 \)
149 \( 1 + T \)
good5 \( 1 - 1.12T + 5T^{2} \)
7 \( 1 + 0.686T + 7T^{2} \)
11 \( 1 + 3.05T + 11T^{2} \)
13 \( 1 + 3.99T + 13T^{2} \)
17 \( 1 + 5.80T + 17T^{2} \)
19 \( 1 - 8.32T + 19T^{2} \)
23 \( 1 - 2.20T + 23T^{2} \)
29 \( 1 + 6.02T + 29T^{2} \)
31 \( 1 - 8.16T + 31T^{2} \)
37 \( 1 - 4.59T + 37T^{2} \)
41 \( 1 + 5.82T + 41T^{2} \)
43 \( 1 - 0.944T + 43T^{2} \)
47 \( 1 - 6.19T + 47T^{2} \)
53 \( 1 - 8.18T + 53T^{2} \)
59 \( 1 + 8.39T + 59T^{2} \)
61 \( 1 + 14.1T + 61T^{2} \)
67 \( 1 - 6.33T + 67T^{2} \)
71 \( 1 + 2.82T + 71T^{2} \)
73 \( 1 + 3.33T + 73T^{2} \)
79 \( 1 + 8.49T + 79T^{2} \)
83 \( 1 + 6.34T + 83T^{2} \)
89 \( 1 + 6.07T + 89T^{2} \)
97 \( 1 + 11.5T + 97T^{2} \)
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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

−7.43970864711213395084349431130, −6.74489815685244030740476955918, −5.96903184182939777579952468927, −5.31305624640855346992313067327, −4.80733420809220822730469496129, −3.99457697646388290797457627628, −2.87034583836617110304461496903, −2.57219669026795524313442132060, −1.48620560088487480092757346853, 0, 1.48620560088487480092757346853, 2.57219669026795524313442132060, 2.87034583836617110304461496903, 3.99457697646388290797457627628, 4.80733420809220822730469496129, 5.31305624640855346992313067327, 5.96903184182939777579952468927, 6.74489815685244030740476955918, 7.43970864711213395084349431130

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