↑

Properties

Label 2-1617-231.131-c0-0-5
Degree $2$
Conductor $1617$
Sign $0.832 - 0.553i$
Analytic cond. $0.806988$
Root an. cond. $0.898325$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + 0.999·6-s + 8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s − 0.999·15-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s − 0.999·22-s + (0.500 + 0.866i)24-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)2-s + (0.5 + 0.866i)3-s + (−0.5 + 0.866i)5-s + 0.999·6-s + 8-s + (−0.499 + 0.866i)9-s + (0.499 + 0.866i)10-s + (−0.5 − 0.866i)11-s + 13-s − 0.999·15-s + (0.5 − 0.866i)16-s + (0.499 + 0.866i)18-s + (−0.5 + 0.866i)19-s − 0.999·22-s + (0.500 + 0.866i)24-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1617 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.832 - 0.553i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1617\)    =    \(3 \cdot 7^{2} \cdot 11\)
Sign: $0.832 - 0.553i$
Analytic conductor: \(0.806988\)
Root analytic conductor: \(0.898325\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1617} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1617,\ (\ :0),\ 0.832 - 0.553i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.694840895\)
\(L(\frac12)\) \(\approx\) \(1.694840895\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 \)
11 \( 1 + (0.5 + 0.866i)T \)
good2 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
13 \( 1 - T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T + T^{2} \)
31 \( 1 + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.5 + 0.866i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 - 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
61 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - T^{2} \)
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

−9.966803099611580266710387753105, −8.916408792717182694782614444226, −8.056494596432547773333560158757, −7.56631002812532037092035267711, −6.30359310396941664586422783680, −5.33699625010765804593622740985, −4.19980800324129698217569904607, −3.53398473171418188034077746532, −3.07739438181177528840837118793, −1.97341314005019756936724903593, 1.19758150455624680786336143199, 2.34803030628735925402772456232, 3.82126033505228320754453969733, 4.66007049095421410557943988879, 5.51337709265343237352831018723, 6.41867839149309715888623461711, 7.08138910915198032487073521748, 7.82550146086057866809733294866, 8.432541596926031692845169630967, 9.162837996421236148024725137701

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