Properties

Label 2-8016-1.1-c1-0-59
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 + 2.77·5-s − 1.86·7-s + 9-s − 2.28·13-s + 2.77·15-s + 3.65·17-s − 4.84·19-s − 1.86·21-s + 4.84·23-s + 2.67·25-s + 27-s + 0.459·29-s + 6.70·31-s − 5.16·35-s + 9.26·37-s − 2.28·39-s − 6.03·41-s + 2.63·43-s + 2.77·45-s − 0.325·47-s − 3.51·49-s + 3.65·51-s + 7.61·53-s − 4.84·57-s − 7.69·59-s + 3.54·61-s + ⋯
L(s)  = 1  + 0.577·3-s + 1.23·5-s − 0.705·7-s + 0.333·9-s − 0.634·13-s + 0.715·15-s + 0.885·17-s − 1.11·19-s − 0.407·21-s + 1.01·23-s + 0.534·25-s + 0.192·27-s + 0.0852·29-s + 1.20·31-s − 0.873·35-s + 1.52·37-s − 0.366·39-s − 0.942·41-s + 0.402·43-s + 0.412·45-s − 0.0474·47-s − 0.502·49-s + 0.511·51-s + 1.04·53-s − 0.641·57-s − 1.00·59-s + 0.453·61-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

\(L(1)\) \(\approx\) \(3.107204039\)
\(L(\frac12)\) \(\approx\) \(3.107204039\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.77T + 5T^{2} \)
7 \( 1 + 1.86T + 7T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + 2.28T + 13T^{2} \)
17 \( 1 - 3.65T + 17T^{2} \)
19 \( 1 + 4.84T + 19T^{2} \)
23 \( 1 - 4.84T + 23T^{2} \)
29 \( 1 - 0.459T + 29T^{2} \)
31 \( 1 - 6.70T + 31T^{2} \)
37 \( 1 - 9.26T + 37T^{2} \)
41 \( 1 + 6.03T + 41T^{2} \)
43 \( 1 - 2.63T + 43T^{2} \)
47 \( 1 + 0.325T + 47T^{2} \)
53 \( 1 - 7.61T + 53T^{2} \)
59 \( 1 + 7.69T + 59T^{2} \)
61 \( 1 - 3.54T + 61T^{2} \)
67 \( 1 - 11.1T + 67T^{2} \)
71 \( 1 + 8.14T + 71T^{2} \)
73 \( 1 - 12.1T + 73T^{2} \)
79 \( 1 + 4.73T + 79T^{2} \)
83 \( 1 - 5.50T + 83T^{2} \)
89 \( 1 - 7.16T + 89T^{2} \)
97 \( 1 + 3.21T + 97T^{2} \)
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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

−7.924550108947715242860190407525, −7.01815051847016763485183568911, −6.46609084824925230558524417220, −5.85625346919624759916420120270, −5.06408587302623030953624558931, −4.30939205241149376849104076150, −3.28167921675390377223356960501, −2.64371969057142825289568650426, −1.96185677486504667243065855093, −0.851341147501869465971857715851, 0.851341147501869465971857715851, 1.96185677486504667243065855093, 2.64371969057142825289568650426, 3.28167921675390377223356960501, 4.30939205241149376849104076150, 5.06408587302623030953624558931, 5.85625346919624759916420120270, 6.46609084824925230558524417220, 7.01815051847016763485183568911, 7.924550108947715242860190407525

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