Properties

Label 2-8046-1.1-c1-0-94
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 − 1.41·5-s − 1.27·7-s − 8-s + 1.41·10-s − 5.29·11-s + 6.20·13-s + 1.27·14-s + 16-s − 6.19·17-s + 4.29·19-s − 1.41·20-s + 5.29·22-s + 6.65·23-s − 2.99·25-s − 6.20·26-s − 1.27·28-s − 4.65·29-s + 0.972·31-s − 32-s + 6.19·34-s + 1.80·35-s + 0.986·37-s − 4.29·38-s + 1.41·40-s + 8.34·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.633·5-s − 0.482·7-s − 0.353·8-s + 0.448·10-s − 1.59·11-s + 1.72·13-s + 0.340·14-s + 0.250·16-s − 1.50·17-s + 0.984·19-s − 0.316·20-s + 1.12·22-s + 1.38·23-s − 0.598·25-s − 1.21·26-s − 0.241·28-s − 0.864·29-s + 0.174·31-s − 0.176·32-s + 1.06·34-s + 0.305·35-s + 0.162·37-s − 0.696·38-s + 0.224·40-s + 1.30·41-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 \( 1 + 1.41T + 5T^{2} \)
7 \( 1 + 1.27T + 7T^{2} \)
11 \( 1 + 5.29T + 11T^{2} \)
13 \( 1 - 6.20T + 13T^{2} \)
17 \( 1 + 6.19T + 17T^{2} \)
19 \( 1 - 4.29T + 19T^{2} \)
23 \( 1 - 6.65T + 23T^{2} \)
29 \( 1 + 4.65T + 29T^{2} \)
31 \( 1 - 0.972T + 31T^{2} \)
37 \( 1 - 0.986T + 37T^{2} \)
41 \( 1 - 8.34T + 41T^{2} \)
43 \( 1 - 3.52T + 43T^{2} \)
47 \( 1 + 4.00T + 47T^{2} \)
53 \( 1 - 0.663T + 53T^{2} \)
59 \( 1 - 4.27T + 59T^{2} \)
61 \( 1 - 3.90T + 61T^{2} \)
67 \( 1 - 2.70T + 67T^{2} \)
71 \( 1 - 2.68T + 71T^{2} \)
73 \( 1 + 4.49T + 73T^{2} \)
79 \( 1 + 5.09T + 79T^{2} \)
83 \( 1 + 11.5T + 83T^{2} \)
89 \( 1 - 8.01T + 89T^{2} \)
97 \( 1 + 2.43T + 97T^{2} \)
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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

−7.56294995532587609001734253875, −7.00045828638953358656030654718, −6.17183818926506125337250618459, −5.55917505571155152290579326557, −4.65305719581592870086408882928, −3.71947337497233594365799831246, −3.05239163515661845728978271170, −2.22906556444728703537734842079, −1.01297343551217573691322013387, 0, 1.01297343551217573691322013387, 2.22906556444728703537734842079, 3.05239163515661845728978271170, 3.71947337497233594365799831246, 4.65305719581592870086408882928, 5.55917505571155152290579326557, 6.17183818926506125337250618459, 7.00045828638953358656030654718, 7.56294995532587609001734253875

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