Properties

Label 2-2016-1.1-c1-0-4
Degree $2$
Conductor $2016$
Sign $1$
Analytic cond. $16.0978$
Root an. cond. $4.01221$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3.23·5-s + 7-s + 6.47·11-s + 0.763·13-s − 4.47·17-s − 1.23·19-s − 4·23-s + 5.47·25-s + 4.47·29-s − 2.47·31-s − 3.23·35-s − 4.47·37-s + 8.47·41-s + 6.47·43-s − 10.4·47-s + 49-s + 10·53-s − 20.9·55-s + 9.23·59-s + 11.2·61-s − 2.47·65-s + 4·67-s + 4.94·71-s − 2.94·73-s + 6.47·77-s + 12.9·79-s + 9.23·83-s + ⋯
L(s)  = 1  − 1.44·5-s + 0.377·7-s + 1.95·11-s + 0.211·13-s − 1.08·17-s − 0.283·19-s − 0.834·23-s + 1.09·25-s + 0.830·29-s − 0.444·31-s − 0.546·35-s − 0.735·37-s + 1.32·41-s + 0.986·43-s − 1.52·47-s + 0.142·49-s + 1.37·53-s − 2.82·55-s + 1.20·59-s + 1.43·61-s − 0.306·65-s + 0.488·67-s + 0.586·71-s − 0.344·73-s + 0.737·77-s + 1.45·79-s + 1.01·83-s + ⋯

Functional equation

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

Invariants

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

Particular Values

\(L(1)\) \(\approx\) \(1.418123791\)
\(L(\frac12)\) \(\approx\) \(1.418123791\)
\(L(\frac{3}{2})\) 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 \( 1 - T \)
good5 \( 1 + 3.23T + 5T^{2} \)
11 \( 1 - 6.47T + 11T^{2} \)
13 \( 1 - 0.763T + 13T^{2} \)
17 \( 1 + 4.47T + 17T^{2} \)
19 \( 1 + 1.23T + 19T^{2} \)
23 \( 1 + 4T + 23T^{2} \)
29 \( 1 - 4.47T + 29T^{2} \)
31 \( 1 + 2.47T + 31T^{2} \)
37 \( 1 + 4.47T + 37T^{2} \)
41 \( 1 - 8.47T + 41T^{2} \)
43 \( 1 - 6.47T + 43T^{2} \)
47 \( 1 + 10.4T + 47T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 - 9.23T + 59T^{2} \)
61 \( 1 - 11.2T + 61T^{2} \)
67 \( 1 - 4T + 67T^{2} \)
71 \( 1 - 4.94T + 71T^{2} \)
73 \( 1 + 2.94T + 73T^{2} \)
79 \( 1 - 12.9T + 79T^{2} \)
83 \( 1 - 9.23T + 83T^{2} \)
89 \( 1 - 6T + 89T^{2} \)
97 \( 1 - 12.4T + 97T^{2} \)
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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

−8.868912064327284131099737566901, −8.499163110453923095836011900585, −7.60215299019775738553685353783, −6.82380813641189119428851670999, −6.21199137994958596205717062655, −4.87151952150274988598379958034, −3.98514069108880664016473976885, −3.74041176419834802253792063886, −2.16371368149281105888051700463, −0.807766492554902541597829732893, 0.807766492554902541597829732893, 2.16371368149281105888051700463, 3.74041176419834802253792063886, 3.98514069108880664016473976885, 4.87151952150274988598379958034, 6.21199137994958596205717062655, 6.82380813641189119428851670999, 7.60215299019775738553685353783, 8.499163110453923095836011900585, 8.868912064327284131099737566901

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