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Properties

Label 2-2016-56.27-c1-0-37
Degree $2$
Conductor $2016$
Sign $-0.935 - 0.353i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.64i·7-s − 4·11-s − 5.29i·23-s − 5·25-s + 10.5i·29-s + 10.5i·37-s − 12·43-s − 7.00·49-s − 10.5i·53-s − 4·67-s + 5.29i·71-s + 10.5i·77-s + 15.8i·79-s − 20·107-s − 10.5i·109-s + ⋯
L(s)  = 1  − 0.999i·7-s − 1.20·11-s − 1.10i·23-s − 25-s + 1.96i·29-s + 1.73i·37-s − 1.82·43-s − 49-s − 1.45i·53-s − 0.488·67-s + 0.627i·71-s + 1.20i·77-s + 1.78i·79-s − 1.93·107-s − 1.01i·109-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.935 - 0.353i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.935 - 0.353i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (559, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(1\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.935 - 0.353i)\)

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 \( 1 + 2.64iT \)
good5 \( 1 + 5T^{2} \)
11 \( 1 + 4T + 11T^{2} \)
13 \( 1 + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 5.29iT - 23T^{2} \)
29 \( 1 - 10.5iT - 29T^{2} \)
31 \( 1 + 31T^{2} \)
37 \( 1 - 10.5iT - 37T^{2} \)
41 \( 1 - 41T^{2} \)
43 \( 1 + 12T + 43T^{2} \)
47 \( 1 + 47T^{2} \)
53 \( 1 + 10.5iT - 53T^{2} \)
59 \( 1 - 59T^{2} \)
61 \( 1 + 61T^{2} \)
67 \( 1 + 4T + 67T^{2} \)
71 \( 1 - 5.29iT - 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 15.8iT - 79T^{2} \)
83 \( 1 - 83T^{2} \)
89 \( 1 - 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−8.452204541761647487291515084816, −8.072616688781879013095660801427, −7.07391546811275128710390715266, −6.55452203787861559976140148927, −5.32869189263723521515890404061, −4.73641998569204157333799592321, −3.67904727443111015087998893816, −2.79537809586371707059243222585, −1.48766920262029554393420409156, 0, 1.90727627342521177906448316218, 2.70383634510207679498960161528, 3.77187694616357225632376612489, 4.89141071780706476996511095553, 5.66905576971278339580661599202, 6.19165019166169781154113114720, 7.52250980811915503025090701641, 7.905009892841643691288199515905, 8.832468448451926565692485463545

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