Properties

Degree $2$
Conductor $8016$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 1.48·5-s + 1.03·7-s + 9-s − 5.64·11-s − 4.47·13-s + 1.48·15-s − 6.13·17-s − 5.87·19-s − 1.03·21-s + 1.22·23-s − 2.80·25-s − 27-s − 6.91·29-s − 3.70·31-s + 5.64·33-s − 1.54·35-s − 6.62·37-s + 4.47·39-s + 1.85·41-s + 2.29·43-s − 1.48·45-s − 3.59·47-s − 5.92·49-s + 6.13·51-s − 0.299·53-s + 8.36·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 0.662·5-s + 0.392·7-s + 0.333·9-s − 1.70·11-s − 1.24·13-s + 0.382·15-s − 1.48·17-s − 1.34·19-s − 0.226·21-s + 0.255·23-s − 0.560·25-s − 0.192·27-s − 1.28·29-s − 0.664·31-s + 0.981·33-s − 0.260·35-s − 1.08·37-s + 0.717·39-s + 0.290·41-s + 0.350·43-s − 0.220·45-s − 0.523·47-s − 0.845·49-s + 0.859·51-s − 0.0411·53-s + 1.12·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8016\)    =    \(2^{4} \cdot 3 \cdot 167\)
Sign: $1$
Motivic weight: \(1\)
Character: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.03506733149\)
\(L(\frac12)\) \(\approx\) \(0.03506733149\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 + 1.48T + 5T^{2} \)
7 \( 1 - 1.03T + 7T^{2} \)
11 \( 1 + 5.64T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 6.13T + 17T^{2} \)
19 \( 1 + 5.87T + 19T^{2} \)
23 \( 1 - 1.22T + 23T^{2} \)
29 \( 1 + 6.91T + 29T^{2} \)
31 \( 1 + 3.70T + 31T^{2} \)
37 \( 1 + 6.62T + 37T^{2} \)
41 \( 1 - 1.85T + 41T^{2} \)
43 \( 1 - 2.29T + 43T^{2} \)
47 \( 1 + 3.59T + 47T^{2} \)
53 \( 1 + 0.299T + 53T^{2} \)
59 \( 1 + 0.102T + 59T^{2} \)
61 \( 1 - 3.64T + 61T^{2} \)
67 \( 1 - 5.27T + 67T^{2} \)
71 \( 1 + 2.00T + 71T^{2} \)
73 \( 1 - 9.34T + 73T^{2} \)
79 \( 1 - 3.92T + 79T^{2} \)
83 \( 1 - 11.4T + 83T^{2} \)
89 \( 1 + 10.4T + 89T^{2} \)
97 \( 1 + 3.40T + 97T^{2} \)
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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

−7.81390513559058933192064746104, −7.17998707158418808318985058312, −6.56953815073944866439787459066, −5.56744152847912423223567248637, −5.02004794859160962983307944490, −4.45997690196027713027103498929, −3.66018235757104397578020207717, −2.44703262183904292016352675582, −1.97358215802997982273294077535, −0.087809471718346019428789993430, 0.087809471718346019428789993430, 1.97358215802997982273294077535, 2.44703262183904292016352675582, 3.66018235757104397578020207717, 4.45997690196027713027103498929, 5.02004794859160962983307944490, 5.56744152847912423223567248637, 6.56953815073944866439787459066, 7.17998707158418808318985058312, 7.81390513559058933192064746104

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