↑

Properties

Label 2-605-5.4-c1-0-19
Degree $2$
Conductor $605$
Sign $0.943 - 0.329i$
Analytic cond. $4.83094$
Root an. cond. $2.19794$
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  + 2.60i·2-s − 2.13i·3-s − 4.80·4-s + (−2.11 + 0.737i)5-s + 5.58·6-s + 0.988i·7-s − 7.30i·8-s − 1.57·9-s + (−1.92 − 5.50i)10-s + 10.2i·12-s − 3.07i·13-s − 2.57·14-s + (1.57 + 4.51i)15-s + 9.44·16-s − 1.61i·17-s − 4.11i·18-s + ⋯
L(s)  = 1  + 1.84i·2-s − 1.23i·3-s − 2.40·4-s + (−0.943 + 0.329i)5-s + 2.27·6-s + 0.373i·7-s − 2.58i·8-s − 0.526·9-s + (−0.608 − 1.74i)10-s + 2.96i·12-s − 0.853i·13-s − 0.689·14-s + (0.407 + 1.16i)15-s + 2.36·16-s − 0.392i·17-s − 0.970i·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.943 - 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.943 - 0.329i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (364, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.943 - 0.329i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.916278 + 0.155521i\)
\(L(\frac12)\) \(\approx\) \(0.916278 + 0.155521i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (2.11 - 0.737i)T \)
11 \( 1 \)
good2 \( 1 - 2.60iT - 2T^{2} \)
3 \( 1 + 2.13iT - 3T^{2} \)
7 \( 1 - 0.988iT - 7T^{2} \)
13 \( 1 + 3.07iT - 13T^{2} \)
17 \( 1 + 1.61iT - 17T^{2} \)
19 \( 1 - 6.58T + 19T^{2} \)
23 \( 1 + 5.18iT - 23T^{2} \)
29 \( 1 - 7.31T + 29T^{2} \)
31 \( 1 + 2.64T + 31T^{2} \)
37 \( 1 - 2.80iT - 37T^{2} \)
41 \( 1 + 1.38T + 41T^{2} \)
43 \( 1 + 3.18iT - 43T^{2} \)
47 \( 1 + 2.13iT - 47T^{2} \)
53 \( 1 - 2.37iT - 53T^{2} \)
59 \( 1 + 12.2T + 59T^{2} \)
61 \( 1 - 1.00T + 61T^{2} \)
67 \( 1 + 9.84iT - 67T^{2} \)
71 \( 1 + 0.243T + 71T^{2} \)
73 \( 1 + 13.6iT - 73T^{2} \)
79 \( 1 - 7.69T + 79T^{2} \)
83 \( 1 - 5.95iT - 83T^{2} \)
89 \( 1 + 9T + 89T^{2} \)
97 \( 1 - 12.7iT - 97T^{2} \)
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

−10.59492488656355414665998217067, −9.355232010386859766560320688600, −8.287161439757323545996693217786, −7.88134040667349341666634086424, −7.11612856997661157268364022770, −6.54494087496294063998720329816, −5.54382975027349778762462158065, −4.54715102319674413676701150999, −3.05806487072659030368895188540, −0.63050858535318676273797077770, 1.26337410015493563998134626053, 3.08535281658664906663450322100, 3.85191574387705217621104433252, 4.46434407551267421905185244881, 5.30245703794715810424241333956, 7.34338376916351061712976541274, 8.525796077844546316247531103261, 9.324620781694908137490400438204, 9.866676043303652242463484776613, 10.71449787621514941504900463954

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