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Properties

Label 2-2016-168.83-c0-0-0
Degree $2$
Conductor $2016$
Sign $0.169 - 0.985i$
Analytic cond. $1.00611$
Root an. cond. $1.00305$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + i·7-s + 1.41i·11-s − 1.41·23-s + 25-s − 1.41·29-s + 2i·37-s − 49-s + 1.41·53-s + 2·67-s + 1.41·71-s − 1.41·77-s − 1.41i·107-s − 2i·109-s + 1.41i·113-s + ⋯
L(s)  = 1  + i·7-s + 1.41i·11-s − 1.41·23-s + 25-s − 1.41·29-s + 2i·37-s − 49-s + 1.41·53-s + 2·67-s + 1.41·71-s − 1.41·77-s − 1.41i·107-s − 2i·109-s + 1.41i·113-s + ⋯

Functional equation

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

Invariants

Degree: \(2\)
Conductor: \(2016\)    =    \(2^{5} \cdot 3^{2} \cdot 7\)
Sign: $0.169 - 0.985i$
Analytic conductor: \(1.00611\)
Root analytic conductor: \(1.00305\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2016} (1007, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2016,\ (\ :0),\ 0.169 - 0.985i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.031135683\)
\(L(\frac12)\) \(\approx\) \(1.031135683\)
\(L(1)\) 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 \( 1 - iT \)
good5 \( 1 - T^{2} \)
11 \( 1 - 1.41iT - T^{2} \)
13 \( 1 + T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + 1.41T + T^{2} \)
29 \( 1 + 1.41T + T^{2} \)
31 \( 1 + T^{2} \)
37 \( 1 - 2iT - T^{2} \)
41 \( 1 + T^{2} \)
43 \( 1 + T^{2} \)
47 \( 1 - T^{2} \)
53 \( 1 - 1.41T + T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 + T^{2} \)
67 \( 1 - 2T + T^{2} \)
71 \( 1 - 1.41T + T^{2} \)
73 \( 1 - T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 + T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−9.663097268876841680990402693796, −8.702447207586259245259064412035, −8.042204817363754871751728043943, −7.13525301828655697145513790571, −6.39587139019178787032935914507, −5.46071527593913837165607330777, −4.75644222286971363959504766446, −3.76577251535822591770078848331, −2.55771722725782783853202071704, −1.75719183807430767758014555908, 0.74474925350974717950481408789, 2.21472478015978211444916988051, 3.55254357999201059416301692300, 3.99191789866107457736242908640, 5.25268386447213208072689829825, 5.97240789573911800497001199288, 6.85552863208009802181910523816, 7.64232388210341663692657419369, 8.339461095978078106402589131648, 9.141065555879891876188866796086

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