Properties

Label 2-8046-1.1-c1-0-39
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 4-s − 2.45·5-s − 3.97·7-s + 8-s − 2.45·10-s + 3.25·11-s + 2.63·13-s − 3.97·14-s + 16-s + 7.31·17-s − 5.34·19-s − 2.45·20-s + 3.25·22-s + 6.94·23-s + 1.05·25-s + 2.63·26-s − 3.97·28-s − 7.73·29-s − 3.43·31-s + 32-s + 7.31·34-s + 9.77·35-s − 5.84·37-s − 5.34·38-s − 2.45·40-s − 9.31·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.10·5-s − 1.50·7-s + 0.353·8-s − 0.777·10-s + 0.981·11-s + 0.730·13-s − 1.06·14-s + 0.250·16-s + 1.77·17-s − 1.22·19-s − 0.550·20-s + 0.694·22-s + 1.44·23-s + 0.210·25-s + 0.516·26-s − 0.751·28-s − 1.43·29-s − 0.616·31-s + 0.176·32-s + 1.25·34-s + 1.65·35-s − 0.961·37-s − 0.866·38-s − 0.388·40-s − 1.45·41-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: \(0\)
Selberg data: \((2,\ 8046,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.073032688\)
\(L(\frac12)\) \(\approx\) \(2.073032688\)
\(L(\frac{3}{2})\) 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 \( 1 + 2.45T + 5T^{2} \)
7 \( 1 + 3.97T + 7T^{2} \)
11 \( 1 - 3.25T + 11T^{2} \)
13 \( 1 - 2.63T + 13T^{2} \)
17 \( 1 - 7.31T + 17T^{2} \)
19 \( 1 + 5.34T + 19T^{2} \)
23 \( 1 - 6.94T + 23T^{2} \)
29 \( 1 + 7.73T + 29T^{2} \)
31 \( 1 + 3.43T + 31T^{2} \)
37 \( 1 + 5.84T + 37T^{2} \)
41 \( 1 + 9.31T + 41T^{2} \)
43 \( 1 + 7.07T + 43T^{2} \)
47 \( 1 - 7.58T + 47T^{2} \)
53 \( 1 - 6.59T + 53T^{2} \)
59 \( 1 + 10.9T + 59T^{2} \)
61 \( 1 - 7.86T + 61T^{2} \)
67 \( 1 - 4.36T + 67T^{2} \)
71 \( 1 + 8.10T + 71T^{2} \)
73 \( 1 - 10.7T + 73T^{2} \)
79 \( 1 - 13.9T + 79T^{2} \)
83 \( 1 - 14.0T + 83T^{2} \)
89 \( 1 + 11.0T + 89T^{2} \)
97 \( 1 - 10.5T + 97T^{2} \)
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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

−7.59572376377654543400570050890, −7.01314106731230060324193864734, −6.48179594623270497245571146470, −5.79136324261308335697862922660, −5.03160528538708761125310707344, −3.92835684147812504543221192903, −3.53833514402871705798648149448, −3.23893964943142900813577950402, −1.83671023490885035928330871462, −0.63490290592735311254655564490, 0.63490290592735311254655564490, 1.83671023490885035928330871462, 3.23893964943142900813577950402, 3.53833514402871705798648149448, 3.92835684147812504543221192903, 5.03160528538708761125310707344, 5.79136324261308335697862922660, 6.48179594623270497245571146470, 7.01314106731230060324193864734, 7.59572376377654543400570050890

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