Properties

Label 2-8046-1.1-c1-0-189
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 + 3.46·5-s − 3.72·7-s + 8-s + 3.46·10-s − 5.41·11-s + 5.45·13-s − 3.72·14-s + 16-s − 5.94·17-s + 5.41·19-s + 3.46·20-s − 5.41·22-s − 4.72·23-s + 6.99·25-s + 5.45·26-s − 3.72·28-s − 5.68·29-s + 4.89·31-s + 32-s − 5.94·34-s − 12.8·35-s − 10.8·37-s + 5.41·38-s + 3.46·40-s − 10.6·41-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.5·4-s + 1.54·5-s − 1.40·7-s + 0.353·8-s + 1.09·10-s − 1.63·11-s + 1.51·13-s − 0.995·14-s + 0.250·16-s − 1.44·17-s + 1.24·19-s + 0.774·20-s − 1.15·22-s − 0.984·23-s + 1.39·25-s + 1.06·26-s − 0.703·28-s − 1.05·29-s + 0.879·31-s + 0.176·32-s − 1.01·34-s − 2.18·35-s − 1.77·37-s + 0.877·38-s + 0.547·40-s − 1.67·41-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 \( 1 - 3.46T + 5T^{2} \)
7 \( 1 + 3.72T + 7T^{2} \)
11 \( 1 + 5.41T + 11T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
17 \( 1 + 5.94T + 17T^{2} \)
19 \( 1 - 5.41T + 19T^{2} \)
23 \( 1 + 4.72T + 23T^{2} \)
29 \( 1 + 5.68T + 29T^{2} \)
31 \( 1 - 4.89T + 31T^{2} \)
37 \( 1 + 10.8T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 + 12.8T + 43T^{2} \)
47 \( 1 + 8.22T + 47T^{2} \)
53 \( 1 - 11.8T + 53T^{2} \)
59 \( 1 + 6.31T + 59T^{2} \)
61 \( 1 - 10.9T + 61T^{2} \)
67 \( 1 + 8.30T + 67T^{2} \)
71 \( 1 + 0.883T + 71T^{2} \)
73 \( 1 - 6.87T + 73T^{2} \)
79 \( 1 - 2.27T + 79T^{2} \)
83 \( 1 + 4.53T + 83T^{2} \)
89 \( 1 + 1.46T + 89T^{2} \)
97 \( 1 - 18.1T + 97T^{2} \)
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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

−7.12560968245747128356442335900, −6.54843808817393635093051727684, −6.09468132940443065517890006005, −5.41252657815060417919414221210, −4.97412771024039295396055975277, −3.64934352420185031233379787721, −3.18752760163027422339313219350, −2.31468441003083583489787242320, −1.62673597670332504914616970440, 0, 1.62673597670332504914616970440, 2.31468441003083583489787242320, 3.18752760163027422339313219350, 3.64934352420185031233379787721, 4.97412771024039295396055975277, 5.41252657815060417919414221210, 6.09468132940443065517890006005, 6.54843808817393635093051727684, 7.12560968245747128356442335900

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