Properties

Label 2-2016-12.11-c1-0-19
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

Related objects

Downloads

Learn more

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

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

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

\(L(1)\) \(\approx\) \(1.495509261\)
\(L(\frac12)\) \(\approx\) \(1.495509261\)
\(L(\frac{3}{2})\) 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 \( 1 + 14T + 97T^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−9.175104463484998582447294120007, −8.203704512559838155648909425584, −7.53996416472029429360829836427, −6.48206157586021002695001319933, −5.64739558181282096177381646039, −4.83705498833167358416491705276, −4.22470355973473536538549117117, −2.95393654252162424218046187240, −1.78781076218911119458511871537, −0.56557605447525627040454183607, 1.35896686894928774431925936307, 2.77783392857995085703289123041, 3.32725408469917901749656852368, 4.46101054022409817401880442905, 5.37244848859719355475287577929, 6.45395562642449660141746778741, 7.06454845603752847662683550309, 7.48679082250974263271284539310, 8.704012257788808466940264483272, 9.334171452299861624061307629047

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