Properties

Label 2-8016-1.1-c1-0-110
Degree $2$
Conductor $8016$
Sign $-1$
Analytic cond. $64.0080$
Root an. cond. $8.00050$
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  − 3-s + 2.45·5-s − 2.43·7-s + 9-s − 2.22·11-s − 1.05·13-s − 2.45·15-s + 0.227·17-s + 2.91·19-s + 2.43·21-s − 2.55·23-s + 1.02·25-s − 27-s − 0.104·29-s + 3.77·31-s + 2.22·33-s − 5.98·35-s − 0.875·37-s + 1.05·39-s + 7.02·41-s + 1.10·43-s + 2.45·45-s − 0.498·47-s − 1.06·49-s − 0.227·51-s + 5.62·53-s − 5.47·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.09·5-s − 0.920·7-s + 0.333·9-s − 0.672·11-s − 0.293·13-s − 0.633·15-s + 0.0551·17-s + 0.669·19-s + 0.531·21-s − 0.532·23-s + 0.205·25-s − 0.192·27-s − 0.0193·29-s + 0.677·31-s + 0.388·33-s − 1.01·35-s − 0.143·37-s + 0.169·39-s + 1.09·41-s + 0.168·43-s + 0.366·45-s − 0.0727·47-s − 0.152·49-s − 0.0318·51-s + 0.772·53-s − 0.738·55-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: $-1$
Analytic conductor: \(64.0080\)
Root analytic conductor: \(8.00050\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ -1)\)

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 \( 1 \)
3 \( 1 + T \)
167 \( 1 - T \)
good5 \( 1 - 2.45T + 5T^{2} \)
7 \( 1 + 2.43T + 7T^{2} \)
11 \( 1 + 2.22T + 11T^{2} \)
13 \( 1 + 1.05T + 13T^{2} \)
17 \( 1 - 0.227T + 17T^{2} \)
19 \( 1 - 2.91T + 19T^{2} \)
23 \( 1 + 2.55T + 23T^{2} \)
29 \( 1 + 0.104T + 29T^{2} \)
31 \( 1 - 3.77T + 31T^{2} \)
37 \( 1 + 0.875T + 37T^{2} \)
41 \( 1 - 7.02T + 41T^{2} \)
43 \( 1 - 1.10T + 43T^{2} \)
47 \( 1 + 0.498T + 47T^{2} \)
53 \( 1 - 5.62T + 53T^{2} \)
59 \( 1 - 2.62T + 59T^{2} \)
61 \( 1 - 4.56T + 61T^{2} \)
67 \( 1 + 9.29T + 67T^{2} \)
71 \( 1 + 7.68T + 71T^{2} \)
73 \( 1 + 9.24T + 73T^{2} \)
79 \( 1 - 1.46T + 79T^{2} \)
83 \( 1 - 7.40T + 83T^{2} \)
89 \( 1 + 1.00T + 89T^{2} \)
97 \( 1 - 4.09T + 97T^{2} \)
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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

−7.37409035679675306193439287977, −6.65338415713537302157083007001, −5.99171487161207508482198131229, −5.59315964067262762913075402650, −4.85324563062329663255589563420, −3.94272305776270274935276560549, −2.93787055977163865361775028138, −2.29778650659006345349623445344, −1.20917530742033982658048586284, 0, 1.20917530742033982658048586284, 2.29778650659006345349623445344, 2.93787055977163865361775028138, 3.94272305776270274935276560549, 4.85324563062329663255589563420, 5.59315964067262762913075402650, 5.99171487161207508482198131229, 6.65338415713537302157083007001, 7.37409035679675306193439287977

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