Properties

Label 2-8046-1.1-c1-0-14
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 − 0.316·5-s − 2.35·7-s − 8-s + 0.316·10-s + 1.38·11-s − 5.41·13-s + 2.35·14-s + 16-s − 4.25·17-s − 4.17·19-s − 0.316·20-s − 1.38·22-s + 6.49·23-s − 4.89·25-s + 5.41·26-s − 2.35·28-s − 0.574·29-s + 5.76·31-s − 32-s + 4.25·34-s + 0.745·35-s − 0.814·37-s + 4.17·38-s + 0.316·40-s − 8.31·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.141·5-s − 0.891·7-s − 0.353·8-s + 0.100·10-s + 0.418·11-s − 1.50·13-s + 0.630·14-s + 0.250·16-s − 1.03·17-s − 0.958·19-s − 0.0707·20-s − 0.295·22-s + 1.35·23-s − 0.979·25-s + 1.06·26-s − 0.445·28-s − 0.106·29-s + 1.03·31-s − 0.176·32-s + 0.729·34-s + 0.126·35-s − 0.133·37-s + 0.677·38-s + 0.0500·40-s − 1.29·41-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

\(L(1)\) \(\approx\) \(0.5262802048\)
\(L(\frac12)\) \(\approx\) \(0.5262802048\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.316T + 5T^{2} \)
7 \( 1 + 2.35T + 7T^{2} \)
11 \( 1 - 1.38T + 11T^{2} \)
13 \( 1 + 5.41T + 13T^{2} \)
17 \( 1 + 4.25T + 17T^{2} \)
19 \( 1 + 4.17T + 19T^{2} \)
23 \( 1 - 6.49T + 23T^{2} \)
29 \( 1 + 0.574T + 29T^{2} \)
31 \( 1 - 5.76T + 31T^{2} \)
37 \( 1 + 0.814T + 37T^{2} \)
41 \( 1 + 8.31T + 41T^{2} \)
43 \( 1 - 3.92T + 43T^{2} \)
47 \( 1 + 4.19T + 47T^{2} \)
53 \( 1 + 7.98T + 53T^{2} \)
59 \( 1 + 12.5T + 59T^{2} \)
61 \( 1 - 11.8T + 61T^{2} \)
67 \( 1 - 11.4T + 67T^{2} \)
71 \( 1 + 11.4T + 71T^{2} \)
73 \( 1 - 9.98T + 73T^{2} \)
79 \( 1 - 4.84T + 79T^{2} \)
83 \( 1 + 14.7T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 + 19.0T + 97T^{2} \)
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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

−7.88817767565235534354002379242, −6.96449061435802766640488690540, −6.75635590111151530096359865458, −6.00016836836649377483225708621, −4.96925850528962208028272827809, −4.32775480669004492352681006161, −3.30365520446073402778273048494, −2.59052559177288747969499868414, −1.77143161716908533344263272542, −0.38042540909278742453760484987, 0.38042540909278742453760484987, 1.77143161716908533344263272542, 2.59052559177288747969499868414, 3.30365520446073402778273048494, 4.32775480669004492352681006161, 4.96925850528962208028272827809, 6.00016836836649377483225708621, 6.75635590111151530096359865458, 6.96449061435802766640488690540, 7.88817767565235534354002379242

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