Properties

Label 2-8046-1.1-c1-0-171
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 + 3.80·5-s + 0.515·7-s − 8-s − 3.80·10-s + 0.940·11-s − 2.83·13-s − 0.515·14-s + 16-s − 3.93·17-s − 3.57·19-s + 3.80·20-s − 0.940·22-s − 1.39·23-s + 9.49·25-s + 2.83·26-s + 0.515·28-s − 8.36·29-s + 2.17·31-s − 32-s + 3.93·34-s + 1.96·35-s + 3.67·37-s + 3.57·38-s − 3.80·40-s − 9.56·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s + 1.70·5-s + 0.194·7-s − 0.353·8-s − 1.20·10-s + 0.283·11-s − 0.786·13-s − 0.137·14-s + 0.250·16-s − 0.954·17-s − 0.819·19-s + 0.851·20-s − 0.200·22-s − 0.290·23-s + 1.89·25-s + 0.556·26-s + 0.0974·28-s − 1.55·29-s + 0.390·31-s − 0.176·32-s + 0.674·34-s + 0.331·35-s + 0.603·37-s + 0.579·38-s − 0.602·40-s − 1.49·41-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 \( 1 - 3.80T + 5T^{2} \)
7 \( 1 - 0.515T + 7T^{2} \)
11 \( 1 - 0.940T + 11T^{2} \)
13 \( 1 + 2.83T + 13T^{2} \)
17 \( 1 + 3.93T + 17T^{2} \)
19 \( 1 + 3.57T + 19T^{2} \)
23 \( 1 + 1.39T + 23T^{2} \)
29 \( 1 + 8.36T + 29T^{2} \)
31 \( 1 - 2.17T + 31T^{2} \)
37 \( 1 - 3.67T + 37T^{2} \)
41 \( 1 + 9.56T + 41T^{2} \)
43 \( 1 + 9.00T + 43T^{2} \)
47 \( 1 - 2.89T + 47T^{2} \)
53 \( 1 - 1.17T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 2.12T + 61T^{2} \)
67 \( 1 + 10.2T + 67T^{2} \)
71 \( 1 - 2.35T + 71T^{2} \)
73 \( 1 - 10.1T + 73T^{2} \)
79 \( 1 + 5.14T + 79T^{2} \)
83 \( 1 + 3.57T + 83T^{2} \)
89 \( 1 + 0.00223T + 89T^{2} \)
97 \( 1 - 3.82T + 97T^{2} \)
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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

−7.45579433980297824950250926327, −6.64009443455139009170522884481, −6.35935373076541619459939649458, −5.45230457654804706705835740732, −4.91245074820023656672407716506, −3.85809955745779153074433109430, −2.66197754371092572683679583416, −2.07138981922815814843856302667, −1.47988293267986527200619469544, 0, 1.47988293267986527200619469544, 2.07138981922815814843856302667, 2.66197754371092572683679583416, 3.85809955745779153074433109430, 4.91245074820023656672407716506, 5.45230457654804706705835740732, 6.35935373076541619459939649458, 6.64009443455139009170522884481, 7.45579433980297824950250926327

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