Properties

Label 2-39e2-13.12-c1-0-52
Degree $2$
Conductor $1521$
Sign $-0.832 - 0.554i$
Analytic cond. $12.1452$
Root an. cond. $3.48500$
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  − 1.56i·2-s − 0.438·4-s − 3.56i·5-s + 0.561i·7-s − 2.43i·8-s − 5.56·10-s + 2i·11-s + 0.876·14-s − 4.68·16-s − 1.56·17-s − 7.12i·19-s + 1.56i·20-s + 3.12·22-s + 2·23-s − 7.68·25-s + ⋯
L(s)  = 1  − 1.10i·2-s − 0.219·4-s − 1.59i·5-s + 0.212i·7-s − 0.862i·8-s − 1.75·10-s + 0.603i·11-s + 0.234·14-s − 1.17·16-s − 0.378·17-s − 1.63i·19-s + 0.349i·20-s + 0.665·22-s + 0.417·23-s − 1.53·25-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1521 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.832 - 0.554i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1521\)    =    \(3^{2} \cdot 13^{2}\)
Sign: $-0.832 - 0.554i$
Analytic conductor: \(12.1452\)
Root analytic conductor: \(3.48500\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1521} (1351, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1521,\ (\ :1/2),\ -0.832 - 0.554i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.388933740\)
\(L(\frac12)\) \(\approx\) \(1.388933740\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
13 \( 1 \)
good2 \( 1 + 1.56iT - 2T^{2} \)
5 \( 1 + 3.56iT - 5T^{2} \)
7 \( 1 - 0.561iT - 7T^{2} \)
11 \( 1 - 2iT - 11T^{2} \)
17 \( 1 + 1.56T + 17T^{2} \)
19 \( 1 + 7.12iT - 19T^{2} \)
23 \( 1 - 2T + 23T^{2} \)
29 \( 1 + 6.68T + 29T^{2} \)
31 \( 1 + 2.56iT - 31T^{2} \)
37 \( 1 - 7.56iT - 37T^{2} \)
41 \( 1 + 1.56iT - 41T^{2} \)
43 \( 1 + 4.56T + 43T^{2} \)
47 \( 1 + 8.24iT - 47T^{2} \)
53 \( 1 - 0.684T + 53T^{2} \)
59 \( 1 - 2.87iT - 59T^{2} \)
61 \( 1 - 3.87T + 61T^{2} \)
67 \( 1 + 4.56iT - 67T^{2} \)
71 \( 1 - 14iT - 71T^{2} \)
73 \( 1 + 10.1iT - 73T^{2} \)
79 \( 1 - 5.43T + 79T^{2} \)
83 \( 1 + 0.876iT - 83T^{2} \)
89 \( 1 + 4.87iT - 89T^{2} \)
97 \( 1 - 8.56iT - 97T^{2} \)
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

−9.142316132817521041099812793427, −8.589651936601402443002036017908, −7.43226082500921078022016950993, −6.60135736738156978987578188732, −5.32425178048999822647102106585, −4.67052590880792840773830734182, −3.83474350001262017664087974199, −2.56165624439918345731330092105, −1.64666864131275140311928411999, −0.52022721083876749509999785499, 2.00553202849314730377912392907, 3.09301942385086955850903871728, 3.98139402152287830611797281540, 5.43106179969596408526065099035, 6.06467410282436298617906518941, 6.73343977855231808573855731968, 7.42707564563321824271228809990, 7.975853888016240526665342666786, 8.938244882488699543498362840123, 9.966469536346010573688524610827

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