↑

Properties

Label 2-1512-24.11-c1-0-84
Degree $2$
Conductor $1512$
Sign $0.675 + 0.737i$
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  + (0.985 + 1.01i)2-s + (−0.0577 + 1.99i)4-s − 0.206·5-s − i·7-s + (−2.08 + 1.91i)8-s + (−0.203 − 0.209i)10-s − 6.42i·11-s − 3.52i·13-s + (1.01 − 0.985i)14-s + (−3.99 − 0.230i)16-s − 3.90i·17-s − 5.48·19-s + (0.0119 − 0.413i)20-s + (6.51 − 6.32i)22-s + 0.880·23-s + ⋯
L(s)  = 1  + (0.696 + 0.717i)2-s + (−0.0288 + 0.999i)4-s − 0.0925·5-s − 0.377i·7-s + (−0.737 + 0.675i)8-s + (−0.0645 − 0.0663i)10-s − 1.93i·11-s − 0.977i·13-s + (0.271 − 0.263i)14-s + (−0.998 − 0.0577i)16-s − 0.947i·17-s − 1.25·19-s + (0.00267 − 0.0925i)20-s + (1.38 − 1.34i)22-s + 0.183·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.675 + 0.737i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1512\)    =    \(2^{3} \cdot 3^{3} \cdot 7\)
Sign: $0.675 + 0.737i$
Analytic conductor: \(12.0733\)
Root analytic conductor: \(3.47467\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1512} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1512,\ (\ :1/2),\ 0.675 + 0.737i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.613282461\)
\(L(\frac12)\) \(\approx\) \(1.613282461\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.985 - 1.01i)T \)
3 \( 1 \)
7 \( 1 + iT \)
good5 \( 1 + 0.206T + 5T^{2} \)
11 \( 1 + 6.42iT - 11T^{2} \)
13 \( 1 + 3.52iT - 13T^{2} \)
17 \( 1 + 3.90iT - 17T^{2} \)
19 \( 1 + 5.48T + 19T^{2} \)
23 \( 1 - 0.880T + 23T^{2} \)
29 \( 1 - 3.20T + 29T^{2} \)
31 \( 1 + 0.0631iT - 31T^{2} \)
37 \( 1 + 9.91iT - 37T^{2} \)
41 \( 1 - 5.94iT - 41T^{2} \)
43 \( 1 + 1.13T + 43T^{2} \)
47 \( 1 + 6.81T + 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 4.23iT - 59T^{2} \)
61 \( 1 - 12.3iT - 61T^{2} \)
67 \( 1 + 5.74T + 67T^{2} \)
71 \( 1 - 10.1T + 71T^{2} \)
73 \( 1 - 8.27T + 73T^{2} \)
79 \( 1 - 4.86iT - 79T^{2} \)
83 \( 1 + 7.51iT - 83T^{2} \)
89 \( 1 + 4.44iT - 89T^{2} \)
97 \( 1 + 17.3T + 97T^{2} \)
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

−9.027663581970052627678401854476, −8.379068376909884611629448468979, −7.76414304774468844394080687954, −6.84073331385851648234884162698, −5.96329291218667663467227424656, −5.43862971231185558711404191852, −4.30783324075287846504027074632, −3.46334532245586943755538526501, −2.63054167445201705891523553393, −0.47702895586334582171779486263, 1.75731368287862317840765219083, 2.27481956842509452053844679011, 3.75289197882252530931449171597, 4.44555562410898595088566189407, 5.14696120344775521013592209289, 6.39490411228620320242350523964, 6.79253187806031003384846103635, 8.054292372625616658026085029166, 9.007376909488873945860237730539, 9.816842970227130735954723451018

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