Properties

Label 2-2016-1.1-c1-0-25
Degree $2$
Conductor $2016$
Sign $-1$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s − 2·11-s − 2·13-s − 4·17-s + 4·19-s − 6·23-s − 5·25-s + 2·29-s − 6·37-s − 8·41-s + 8·43-s − 4·47-s + 49-s + 6·53-s − 14·61-s − 4·67-s − 2·71-s − 2·73-s − 2·77-s − 4·79-s + 12·83-s − 2·91-s + 6·97-s − 12·101-s + 8·103-s + 6·107-s − 18·109-s + ⋯
L(s)  = 1  + 0.377·7-s − 0.603·11-s − 0.554·13-s − 0.970·17-s + 0.917·19-s − 1.25·23-s − 25-s + 0.371·29-s − 0.986·37-s − 1.24·41-s + 1.21·43-s − 0.583·47-s + 1/7·49-s + 0.824·53-s − 1.79·61-s − 0.488·67-s − 0.237·71-s − 0.234·73-s − 0.227·77-s − 0.450·79-s + 1.31·83-s − 0.209·91-s + 0.609·97-s − 1.19·101-s + 0.788·103-s + 0.580·107-s − 1.72·109-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: yes
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 \( 1 + p T^{2} \)
11 \( 1 + 2 T + p T^{2} \)
13 \( 1 + 2 T + p T^{2} \)
17 \( 1 + 4 T + p T^{2} \)
19 \( 1 - 4 T + p T^{2} \)
23 \( 1 + 6 T + p T^{2} \)
29 \( 1 - 2 T + p T^{2} \)
31 \( 1 + p T^{2} \)
37 \( 1 + 6 T + p T^{2} \)
41 \( 1 + 8 T + p T^{2} \)
43 \( 1 - 8 T + p T^{2} \)
47 \( 1 + 4 T + p T^{2} \)
53 \( 1 - 6 T + p T^{2} \)
59 \( 1 + p T^{2} \)
61 \( 1 + 14 T + p T^{2} \)
67 \( 1 + 4 T + p T^{2} \)
71 \( 1 + 2 T + p T^{2} \)
73 \( 1 + 2 T + p T^{2} \)
79 \( 1 + 4 T + p T^{2} \)
83 \( 1 - 12 T + p T^{2} \)
89 \( 1 + p T^{2} \)
97 \( 1 - 6 T + p T^{2} \)
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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

−8.728465219026174325197401380467, −7.908788854799808135518860558091, −7.35099727767244243949436898839, −6.37197187863691936186639353981, −5.49391588685950886857050871947, −4.74120936441215238109926770183, −3.82735621857672251034800538625, −2.67528968157297083403654922205, −1.72748916226402630622539746394, 0, 1.72748916226402630622539746394, 2.67528968157297083403654922205, 3.82735621857672251034800538625, 4.74120936441215238109926770183, 5.49391588685950886857050871947, 6.37197187863691936186639353981, 7.35099727767244243949436898839, 7.908788854799808135518860558091, 8.728465219026174325197401380467

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