Properties

Label 2-8046-1.1-c1-0-118
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 − 1.51·5-s + 4.35·7-s − 8-s + 1.51·10-s − 5.50·11-s − 0.889·13-s − 4.35·14-s + 16-s + 0.257·17-s − 3.48·19-s − 1.51·20-s + 5.50·22-s + 2.69·23-s − 2.69·25-s + 0.889·26-s + 4.35·28-s + 1.61·29-s − 0.129·31-s − 32-s − 0.257·34-s − 6.60·35-s + 9.62·37-s + 3.48·38-s + 1.51·40-s − 1.32·41-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.5·4-s − 0.679·5-s + 1.64·7-s − 0.353·8-s + 0.480·10-s − 1.66·11-s − 0.246·13-s − 1.16·14-s + 0.250·16-s + 0.0625·17-s − 0.800·19-s − 0.339·20-s + 1.17·22-s + 0.561·23-s − 0.538·25-s + 0.174·26-s + 0.822·28-s + 0.299·29-s − 0.0233·31-s − 0.176·32-s − 0.0442·34-s − 1.11·35-s + 1.58·37-s + 0.565·38-s + 0.240·40-s − 0.206·41-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 \( 1 + 1.51T + 5T^{2} \)
7 \( 1 - 4.35T + 7T^{2} \)
11 \( 1 + 5.50T + 11T^{2} \)
13 \( 1 + 0.889T + 13T^{2} \)
17 \( 1 - 0.257T + 17T^{2} \)
19 \( 1 + 3.48T + 19T^{2} \)
23 \( 1 - 2.69T + 23T^{2} \)
29 \( 1 - 1.61T + 29T^{2} \)
31 \( 1 + 0.129T + 31T^{2} \)
37 \( 1 - 9.62T + 37T^{2} \)
41 \( 1 + 1.32T + 41T^{2} \)
43 \( 1 - 8.65T + 43T^{2} \)
47 \( 1 + 2.28T + 47T^{2} \)
53 \( 1 - 1.25T + 53T^{2} \)
59 \( 1 - 7.39T + 59T^{2} \)
61 \( 1 + 3.49T + 61T^{2} \)
67 \( 1 + 3.45T + 67T^{2} \)
71 \( 1 + 14.5T + 71T^{2} \)
73 \( 1 + 3.70T + 73T^{2} \)
79 \( 1 - 9.40T + 79T^{2} \)
83 \( 1 - 16.5T + 83T^{2} \)
89 \( 1 + 12.8T + 89T^{2} \)
97 \( 1 + 11.8T + 97T^{2} \)
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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

−7.67391912493065236350212724773, −7.21996173177343518783938508669, −6.09671012897091608236827768773, −5.33805109303149275725244578666, −4.69588936020220466653797588207, −4.03580640364690470928430813458, −2.76696841935528784380104048746, −2.22245519356493874201759279132, −1.15008094299643800321177245197, 0, 1.15008094299643800321177245197, 2.22245519356493874201759279132, 2.76696841935528784380104048746, 4.03580640364690470928430813458, 4.69588936020220466653797588207, 5.33805109303149275725244578666, 6.09671012897091608236827768773, 7.21996173177343518783938508669, 7.67391912493065236350212724773

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