Properties

Label 2-2016-1.1-c1-0-26
Degree $2$
Conductor $2016$
Sign $-1$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
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  + 1.23·5-s − 7-s − 1.23·11-s − 4.47·13-s − 5.23·17-s + 6.47·19-s + 7.70·23-s − 3.47·25-s − 10.4·29-s − 6.47·31-s − 1.23·35-s + 4.47·37-s − 2.76·41-s − 4.94·43-s − 10.4·47-s + 49-s − 8·53-s − 1.52·55-s − 2.47·59-s + 4.47·61-s − 5.52·65-s + 8·67-s − 2.76·71-s + 2.94·73-s + 1.23·77-s + 4.94·79-s + 4.94·83-s + ⋯
L(s)  = 1  + 0.552·5-s − 0.377·7-s − 0.372·11-s − 1.24·13-s − 1.26·17-s + 1.48·19-s + 1.60·23-s − 0.694·25-s − 1.94·29-s − 1.16·31-s − 0.208·35-s + 0.735·37-s − 0.431·41-s − 0.753·43-s − 1.52·47-s + 0.142·49-s − 1.09·53-s − 0.206·55-s − 0.321·59-s + 0.572·61-s − 0.685·65-s + 0.977·67-s − 0.328·71-s + 0.344·73-s + 0.140·77-s + 0.556·79-s + 0.542·83-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-1$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -1)\)

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 \( 1 \)
7 \( 1 + T \)
good5 \( 1 - 1.23T + 5T^{2} \)
11 \( 1 + 1.23T + 11T^{2} \)
13 \( 1 + 4.47T + 13T^{2} \)
17 \( 1 + 5.23T + 17T^{2} \)
19 \( 1 - 6.47T + 19T^{2} \)
23 \( 1 - 7.70T + 23T^{2} \)
29 \( 1 + 10.4T + 29T^{2} \)
31 \( 1 + 6.47T + 31T^{2} \)
37 \( 1 - 4.47T + 37T^{2} \)
41 \( 1 + 2.76T + 41T^{2} \)
43 \( 1 + 4.94T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 + 8T + 53T^{2} \)
59 \( 1 + 2.47T + 59T^{2} \)
61 \( 1 - 4.47T + 61T^{2} \)
67 \( 1 - 8T + 67T^{2} \)
71 \( 1 + 2.76T + 71T^{2} \)
73 \( 1 - 2.94T + 73T^{2} \)
79 \( 1 - 4.94T + 79T^{2} \)
83 \( 1 - 4.94T + 83T^{2} \)
89 \( 1 + 15.7T + 89T^{2} \)
97 \( 1 - 10.9T + 97T^{2} \)
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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

−9.058578811452971063714901843593, −7.81305893681966703350187237703, −7.21818812842580051593077613751, −6.46529429570509566969453408719, −5.36634886141793785225601205236, −4.97187918576383382714219265577, −3.65982390800847195502295349777, −2.71917937749607282511954039318, −1.74755497359716902407833926840, 0, 1.74755497359716902407833926840, 2.71917937749607282511954039318, 3.65982390800847195502295349777, 4.97187918576383382714219265577, 5.36634886141793785225601205236, 6.46529429570509566969453408719, 7.21818812842580051593077613751, 7.81305893681966703350187237703, 9.058578811452971063714901843593

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