Properties

Label 2-2016-12.11-c1-0-23
Degree $2$
Conductor $2016$
Sign $-0.985 - 0.169i$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 2.82i·5-s i·7-s − 1.41·11-s − 2·13-s − 7.07·23-s − 3.00·25-s − 9.89i·29-s + 4i·31-s − 2.82·35-s + 2.82i·41-s + 4i·43-s − 2.82·47-s − 49-s − 4.24i·53-s + 4.00i·55-s + ⋯
L(s)  = 1  − 1.26i·5-s − 0.377i·7-s − 0.426·11-s − 0.554·13-s − 1.47·23-s − 0.600·25-s − 1.83i·29-s + 0.718i·31-s − 0.478·35-s + 0.441i·41-s + 0.609i·43-s − 0.412·47-s − 0.142·49-s − 0.582i·53-s + 0.539i·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.985 - 0.169i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.985 - 0.169i$
Analytic conductor: \(16.0978\)
Root analytic conductor: \(4.01221\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (575, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :1/2),\ -0.985 - 0.169i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.5385346126\)
\(L(\frac12)\) \(\approx\) \(0.5385346126\)
\(L(\frac{3}{2})\) 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 \( 1 + iT \)
good5 \( 1 + 2.82iT - 5T^{2} \)
11 \( 1 + 1.41T + 11T^{2} \)
13 \( 1 + 2T + 13T^{2} \)
17 \( 1 - 17T^{2} \)
19 \( 1 - 19T^{2} \)
23 \( 1 + 7.07T + 23T^{2} \)
29 \( 1 + 9.89iT - 29T^{2} \)
31 \( 1 - 4iT - 31T^{2} \)
37 \( 1 + 37T^{2} \)
41 \( 1 - 2.82iT - 41T^{2} \)
43 \( 1 - 4iT - 43T^{2} \)
47 \( 1 + 2.82T + 47T^{2} \)
53 \( 1 + 4.24iT - 53T^{2} \)
59 \( 1 - 2.82T + 59T^{2} \)
61 \( 1 + 14T + 61T^{2} \)
67 \( 1 - 10iT - 67T^{2} \)
71 \( 1 - 7.07T + 71T^{2} \)
73 \( 1 + 6T + 73T^{2} \)
79 \( 1 - 14iT - 79T^{2} \)
83 \( 1 + 11.3T + 83T^{2} \)
89 \( 1 - 8.48iT - 89T^{2} \)
97 \( 1 + 14T + 97T^{2} \)
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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

−8.562388125035950299082070839179, −8.114324781043737924608546505940, −7.33741453208251919975745664479, −6.25456939991234159985501383387, −5.45134145812226616470449838336, −4.61929811004212363961206708831, −4.00924234978420940382886947876, −2.64997537838759852476012929644, −1.47874488893559954389493611207, −0.18336990947785650624021473923, 1.91814148541747434332632901686, 2.80212292510036994662328863548, 3.59958093013743889585048963707, 4.74802571157142289038999913584, 5.70958883213699101794291930542, 6.43441876828900572970375733728, 7.25827716433049915597682272227, 7.82957903175270038131401809991, 8.806786070206760093486480569353, 9.637055315887257287363720956340

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