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Properties

Label 2-2016-56.13-c0-0-1
Degree $2$
Conductor $2016$
Sign $1$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 7-s + 2·23-s − 25-s + 49-s − 2·71-s + 2·79-s + 2·113-s + ⋯
L(s)  = 1  + 7-s + 2·23-s − 25-s + 49-s − 2·71-s + 2·79-s + 2·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $1$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{2016} (433, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321842395\)
\(L(\frac12)\) \(\approx\) \(1.321842395\)
\(L(1)\) 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 \( 1 + T^{2} \)
11 \( ( 1 - T )( 1 + T ) \)
13 \( 1 + T^{2} \)
17 \( ( 1 - T )( 1 + T ) \)
19 \( 1 + T^{2} \)
23 \( ( 1 - T )^{2} \)
29 \( ( 1 - T )( 1 + T ) \)
31 \( ( 1 - T )( 1 + T ) \)
37 \( ( 1 - T )( 1 + T ) \)
41 \( ( 1 - T )( 1 + T ) \)
43 \( ( 1 - T )( 1 + T ) \)
47 \( ( 1 - T )( 1 + T ) \)
53 \( ( 1 - T )( 1 + T ) \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( ( 1 - T )( 1 + T ) \)
71 \( ( 1 + T )^{2} \)
73 \( ( 1 - T )( 1 + T ) \)
79 \( ( 1 - T )^{2} \)
83 \( 1 + T^{2} \)
89 \( ( 1 - T )( 1 + T ) \)
97 \( ( 1 - T )( 1 + T ) \)
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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

−9.196521524049055996965274831445, −8.609845861317562863991428662898, −7.73809760233912939438783173517, −7.15154253672002626812165505831, −6.13778775498451419586192957947, −5.22219344825821505019401964028, −4.59050143743240089575299917476, −3.54015364659302204675424728415, −2.41728989834666271512798557334, −1.27340330965313936869979181495, 1.27340330965313936869979181495, 2.41728989834666271512798557334, 3.54015364659302204675424728415, 4.59050143743240089575299917476, 5.22219344825821505019401964028, 6.13778775498451419586192957947, 7.15154253672002626812165505831, 7.73809760233912939438783173517, 8.609845861317562863991428662898, 9.196521524049055996965274831445

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