Properties

Label 2-8046-1.1-c1-0-42
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 − 3.47·5-s + 2.01·7-s + 8-s − 3.47·10-s − 5.41·11-s + 4.76·13-s + 2.01·14-s + 16-s + 0.975·17-s + 2.00·19-s − 3.47·20-s − 5.41·22-s + 4.93·23-s + 7.10·25-s + 4.76·26-s + 2.01·28-s + 4.34·29-s − 0.676·31-s + 32-s + 0.975·34-s − 7.01·35-s − 8.00·37-s + 2.00·38-s − 3.47·40-s − 5.01·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s − 1.55·5-s + 0.761·7-s + 0.353·8-s − 1.10·10-s − 1.63·11-s + 1.32·13-s + 0.538·14-s + 0.250·16-s + 0.236·17-s + 0.461·19-s − 0.778·20-s − 1.15·22-s + 1.02·23-s + 1.42·25-s + 0.933·26-s + 0.380·28-s + 0.806·29-s − 0.121·31-s + 0.176·32-s + 0.167·34-s − 1.18·35-s − 1.31·37-s + 0.326·38-s − 0.550·40-s − 0.783·41-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

\(L(1)\) \(\approx\) \(2.397433284\)
\(L(\frac12)\) \(\approx\) \(2.397433284\)
\(L(\frac{3}{2})\) 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 \( 1 + 3.47T + 5T^{2} \)
7 \( 1 - 2.01T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 4.76T + 13T^{2} \)
17 \( 1 - 0.975T + 17T^{2} \)
19 \( 1 - 2.00T + 19T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 4.34T + 29T^{2} \)
31 \( 1 + 0.676T + 31T^{2} \)
37 \( 1 + 8.00T + 37T^{2} \)
41 \( 1 + 5.01T + 41T^{2} \)
43 \( 1 + 2.02T + 43T^{2} \)
47 \( 1 + 9.42T + 47T^{2} \)
53 \( 1 + 2.40T + 53T^{2} \)
59 \( 1 - 6.80T + 59T^{2} \)
61 \( 1 - 14.2T + 61T^{2} \)
67 \( 1 + 5.70T + 67T^{2} \)
71 \( 1 + 13.5T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 0.737T + 79T^{2} \)
83 \( 1 - 12.0T + 83T^{2} \)
89 \( 1 + 0.894T + 89T^{2} \)
97 \( 1 + 6.58T + 97T^{2} \)
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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

−7.87453363119047937675220175378, −7.18657077733581569911994357287, −6.53775585275100322331482503345, −5.40861368570834156616441653253, −5.02480374628316516962161421686, −4.35318402653063199216478752490, −3.37428179289641143462517623099, −3.14449395187175332380161893230, −1.84208644952441579919959073309, −0.69015862631680131573830427838, 0.69015862631680131573830427838, 1.84208644952441579919959073309, 3.14449395187175332380161893230, 3.37428179289641143462517623099, 4.35318402653063199216478752490, 5.02480374628316516962161421686, 5.40861368570834156616441653253, 6.53775585275100322331482503345, 7.18657077733581569911994357287, 7.87453363119047937675220175378

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