Properties

Label 2-1512-21.20-c1-0-0
Degree $2$
Conductor $1512$
Sign $-0.999 + 0.0288i$
Analytic cond. $12.0733$
Root an. cond. $3.47467$
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  + 1.10·5-s + (−2.64 + 0.0763i)7-s + 3.42i·11-s + 4.98i·13-s − 6.38·17-s − 4.65i·19-s − 8.98i·23-s − 3.77·25-s + 1.51i·29-s − 6.71i·31-s + (−2.92 + 0.0843i)35-s − 2.83·37-s − 10.1·41-s + 10.8·43-s − 12.8·47-s + ⋯
L(s)  = 1  + 0.494·5-s + (−0.999 + 0.0288i)7-s + 1.03i·11-s + 1.38i·13-s − 1.54·17-s − 1.06i·19-s − 1.87i·23-s − 0.755·25-s + 0.280i·29-s − 1.20i·31-s + (−0.494 + 0.0142i)35-s − 0.465·37-s − 1.57·41-s + 1.66·43-s − 1.88·47-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0288i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $-0.999 + 0.0288i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (377, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ -0.999 + 0.0288i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1395607899\)
\(L(\frac12)\) \(\approx\) \(0.1395607899\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.64 - 0.0763i)T \)
good5 \( 1 - 1.10T + 5T^{2} \)
11 \( 1 - 3.42iT - 11T^{2} \)
13 \( 1 - 4.98iT - 13T^{2} \)
17 \( 1 + 6.38T + 17T^{2} \)
19 \( 1 + 4.65iT - 19T^{2} \)
23 \( 1 + 8.98iT - 23T^{2} \)
29 \( 1 - 1.51iT - 29T^{2} \)
31 \( 1 + 6.71iT - 31T^{2} \)
37 \( 1 + 2.83T + 37T^{2} \)
41 \( 1 + 10.1T + 41T^{2} \)
43 \( 1 - 10.8T + 43T^{2} \)
47 \( 1 + 12.8T + 47T^{2} \)
53 \( 1 - 3.02iT - 53T^{2} \)
59 \( 1 + 9.54T + 59T^{2} \)
61 \( 1 - 9.16iT - 61T^{2} \)
67 \( 1 + 5.07T + 67T^{2} \)
71 \( 1 - 8.94iT - 71T^{2} \)
73 \( 1 - 7.79iT - 73T^{2} \)
79 \( 1 + 2.05T + 79T^{2} \)
83 \( 1 + 5.73T + 83T^{2} \)
89 \( 1 - 6.47T + 89T^{2} \)
97 \( 1 + 8.04iT - 97T^{2} \)
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

−9.705945986603802840939035250895, −9.207636251068585250899016678634, −8.530458484024939120422288150933, −7.09201773188380534798435003536, −6.74221252882532413769191569260, −6.01601376084783209822436052776, −4.64281770188965859490144119943, −4.19863282061014550348328802819, −2.66784003577563732470093627733, −1.96514849567579159497331213560, 0.05015364249702227532163094132, 1.71201607705498305854253225766, 3.12260261885209879617638782513, 3.61131479779956903647859222305, 5.08895838208941644937349201750, 5.90702966154333490846555732134, 6.40558409396280784129891889894, 7.50529240297954039554189422171, 8.316290916184254043476494260202, 9.147272647841733533449356034090

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