Properties

Label 2-8046-1.1-c1-0-85
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 − 3.89·5-s + 0.649·7-s − 8-s + 3.89·10-s − 1.54·11-s + 0.865·13-s − 0.649·14-s + 16-s − 3.39·17-s + 1.84·19-s − 3.89·20-s + 1.54·22-s + 3.92·23-s + 10.1·25-s − 0.865·26-s + 0.649·28-s − 2.02·29-s + 5.05·31-s − 32-s + 3.39·34-s − 2.52·35-s − 5.61·37-s − 1.84·38-s + 3.89·40-s − 3.49·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 1.74·5-s + 0.245·7-s − 0.353·8-s + 1.23·10-s − 0.466·11-s + 0.240·13-s − 0.173·14-s + 0.250·16-s − 0.823·17-s + 0.422·19-s − 0.870·20-s + 0.329·22-s + 0.817·23-s + 2.03·25-s − 0.169·26-s + 0.122·28-s − 0.376·29-s + 0.907·31-s − 0.176·32-s + 0.582·34-s − 0.427·35-s − 0.923·37-s − 0.298·38-s + 0.615·40-s − 0.545·41-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 \( 1 + 3.89T + 5T^{2} \)
7 \( 1 - 0.649T + 7T^{2} \)
11 \( 1 + 1.54T + 11T^{2} \)
13 \( 1 - 0.865T + 13T^{2} \)
17 \( 1 + 3.39T + 17T^{2} \)
19 \( 1 - 1.84T + 19T^{2} \)
23 \( 1 - 3.92T + 23T^{2} \)
29 \( 1 + 2.02T + 29T^{2} \)
31 \( 1 - 5.05T + 31T^{2} \)
37 \( 1 + 5.61T + 37T^{2} \)
41 \( 1 + 3.49T + 41T^{2} \)
43 \( 1 + 4.91T + 43T^{2} \)
47 \( 1 + 9.06T + 47T^{2} \)
53 \( 1 - 13.6T + 53T^{2} \)
59 \( 1 + 0.861T + 59T^{2} \)
61 \( 1 + 6.49T + 61T^{2} \)
67 \( 1 - 10.7T + 67T^{2} \)
71 \( 1 - 0.839T + 71T^{2} \)
73 \( 1 - 9.60T + 73T^{2} \)
79 \( 1 - 11.5T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 - 10.0T + 89T^{2} \)
97 \( 1 - 6.98T + 97T^{2} \)
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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

−7.68858263409065769157248977973, −6.91720599660818263206457269952, −6.50270027330530475125354732398, −5.18222130960825578438308685708, −4.73059579342159111612027495800, −3.70494324650708480757709323416, −3.22020484201118202814615210980, −2.16250019123360919272784145443, −0.945495136341319071146647454673, 0, 0.945495136341319071146647454673, 2.16250019123360919272784145443, 3.22020484201118202814615210980, 3.70494324650708480757709323416, 4.73059579342159111612027495800, 5.18222130960825578438308685708, 6.50270027330530475125354732398, 6.91720599660818263206457269952, 7.68858263409065769157248977973

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