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Properties

Label 2-2016-12.11-c1-0-18
Degree $2$
Conductor $2016$
Sign $-0.169 + 0.985i$
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  + 2.20i·5-s − i·7-s − 0.794·11-s − 5.12·13-s − 0.620i·17-s − 4i·19-s − 0.794·23-s + 0.123·25-s − 3.00i·29-s − 6.24i·31-s + 2.20·35-s − 11.1·37-s − 3.44i·41-s − 4i·43-s + 12.9·47-s + ⋯
L(s)  = 1  + 0.987i·5-s − 0.377i·7-s − 0.239·11-s − 1.42·13-s − 0.150i·17-s − 0.917i·19-s − 0.165·23-s + 0.0246·25-s − 0.557i·29-s − 1.12i·31-s + 0.373·35-s − 1.82·37-s − 0.538i·41-s − 0.609i·43-s + 1.88·47-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $-0.169 + 0.985i$
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: \((2,\ 2016,\ (\ :1/2),\ -0.169 + 0.985i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.7975758944\)
\(L(\frac12)\) \(\approx\) \(0.7975758944\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.20iT - 5T^{2} \)
11 \( 1 + 0.794T + 11T^{2} \)
13 \( 1 + 5.12T + 13T^{2} \)
17 \( 1 + 0.620iT - 17T^{2} \)
19 \( 1 + 4iT - 19T^{2} \)
23 \( 1 + 0.794T + 23T^{2} \)
29 \( 1 + 3.00iT - 29T^{2} \)
31 \( 1 + 6.24iT - 31T^{2} \)
37 \( 1 + 11.1T + 37T^{2} \)
41 \( 1 + 3.44iT - 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 12.9T + 47T^{2} \)
53 \( 1 + 3.00iT - 53T^{2} \)
59 \( 1 + 1.58T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 5.12iT - 67T^{2} \)
71 \( 1 + 8.03T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 - 11.3iT - 79T^{2} \)
83 \( 1 - 14.4T + 83T^{2} \)
89 \( 1 - 2.20iT - 89T^{2} \)
97 \( 1 - 1.12T + 97T^{2} \)
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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

−9.050907983513795800119955866917, −7.985084970686259751457360916841, −7.15914449457623819938269498426, −6.90291581213312466252060297138, −5.74462079029220760075999916326, −4.88854232097078074412507531068, −3.94231821038682456875550473189, −2.87180871797044283591976898921, −2.16996402354540747242378977655, −0.28223686383804975684864475644, 1.33232133003944003509247482378, 2.44925116406517956230520142066, 3.57856495858378030835014443624, 4.76522366971901229998700167833, 5.16491644593716147144844052111, 6.08802675142914383756441449117, 7.15264861787417340135658543534, 7.83932543863185225452381274556, 8.743815375601309211230610777066, 9.155259482993309359808514403830

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