Properties

Label 2-8016-1.1-c1-0-33
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 3-s + 0.416·5-s − 4.65·7-s + 9-s + 0.283·13-s + 0.416·15-s + 0.640·17-s + 5.88·19-s − 4.65·21-s − 5.88·23-s − 4.82·25-s + 27-s + 5.16·29-s − 1.22·31-s − 1.94·35-s − 6.82·37-s + 0.283·39-s + 12.4·41-s + 3.07·43-s + 0.416·45-s − 7.82·47-s + 14.7·49-s + 0.640·51-s − 5.46·53-s + 5.88·57-s − 3.20·59-s − 1.16·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 0.186·5-s − 1.76·7-s + 0.333·9-s + 0.0785·13-s + 0.107·15-s + 0.155·17-s + 1.35·19-s − 1.01·21-s − 1.22·23-s − 0.965·25-s + 0.192·27-s + 0.959·29-s − 0.220·31-s − 0.328·35-s − 1.12·37-s + 0.0453·39-s + 1.93·41-s + 0.469·43-s + 0.0620·45-s − 1.14·47-s + 2.10·49-s + 0.0897·51-s − 0.751·53-s + 0.779·57-s − 0.417·59-s − 0.149·61-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: $\chi_{8016} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8016,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.836817742\)
\(L(\frac12)\) \(\approx\) \(1.836817742\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.416T + 5T^{2} \)
7 \( 1 + 4.65T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 - 0.283T + 13T^{2} \)
17 \( 1 - 0.640T + 17T^{2} \)
19 \( 1 - 5.88T + 19T^{2} \)
23 \( 1 + 5.88T + 23T^{2} \)
29 \( 1 - 5.16T + 29T^{2} \)
31 \( 1 + 1.22T + 31T^{2} \)
37 \( 1 + 6.82T + 37T^{2} \)
41 \( 1 - 12.4T + 41T^{2} \)
43 \( 1 - 3.07T + 43T^{2} \)
47 \( 1 + 7.82T + 47T^{2} \)
53 \( 1 + 5.46T + 53T^{2} \)
59 \( 1 + 3.20T + 59T^{2} \)
61 \( 1 + 1.16T + 61T^{2} \)
67 \( 1 - 2.29T + 67T^{2} \)
71 \( 1 - 8.60T + 71T^{2} \)
73 \( 1 - 2.26T + 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 + 9.27T + 83T^{2} \)
89 \( 1 - 3.94T + 89T^{2} \)
97 \( 1 - 8.99T + 97T^{2} \)
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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

−7.71453985594708889637874880981, −7.25796442730082104047934227281, −6.29856190161680848847217512250, −6.03112279347668988134905783861, −5.06820664799839549207087459323, −4.01610641620631014532253822670, −3.42585427851412239403730044176, −2.83657092399691397297121727945, −1.91429545866810556926623857323, −0.63255404388318323404800543044, 0.63255404388318323404800543044, 1.91429545866810556926623857323, 2.83657092399691397297121727945, 3.42585427851412239403730044176, 4.01610641620631014532253822670, 5.06820664799839549207087459323, 6.03112279347668988134905783861, 6.29856190161680848847217512250, 7.25796442730082104047934227281, 7.71453985594708889637874880981

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