↑

Properties

Label 2-605-11.5-c1-0-11
Degree $2$
Conductor $605$
Sign $-0.522 - 0.852i$
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  + (0.535 + 1.64i)2-s + (0.809 + 0.587i)3-s + (−0.809 + 0.587i)4-s + (−0.309 + 0.951i)5-s + (−0.535 + 1.64i)6-s + (1.40 − 1.01i)7-s + (1.40 + 1.01i)8-s + (−0.618 − 1.90i)9-s − 1.73·10-s − 0.999·12-s + (1.07 + 3.29i)13-s + (2.42 + 1.76i)14-s + (−0.809 + 0.587i)15-s + (−1.54 + 4.75i)16-s + (−2.14 + 6.58i)17-s + (2.80 − 2.03i)18-s + ⋯
L(s)  = 1  + (0.378 + 1.16i)2-s + (0.467 + 0.339i)3-s + (−0.404 + 0.293i)4-s + (−0.138 + 0.425i)5-s + (−0.218 + 0.672i)6-s + (0.529 − 0.384i)7-s + (0.495 + 0.359i)8-s + (−0.206 − 0.634i)9-s − 0.547·10-s − 0.288·12-s + (0.296 + 0.913i)13-s + (0.648 + 0.471i)14-s + (−0.208 + 0.151i)15-s + (−0.386 + 1.18i)16-s + (−0.519 + 1.59i)17-s + (0.660 − 0.479i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.522 - 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.522 - 0.852i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (511, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.522 - 0.852i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10691 + 1.97672i\)
\(L(\frac12)\) \(\approx\) \(1.10691 + 1.97672i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.309 - 0.951i)T \)
11 \( 1 \)
good2 \( 1 + (-0.535 - 1.64i)T + (-1.61 + 1.17i)T^{2} \)
3 \( 1 + (-0.809 - 0.587i)T + (0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.40 + 1.01i)T + (2.16 - 6.65i)T^{2} \)
13 \( 1 + (-1.07 - 3.29i)T + (-10.5 + 7.64i)T^{2} \)
17 \( 1 + (2.14 - 6.58i)T + (-13.7 - 9.99i)T^{2} \)
19 \( 1 + (-2.80 - 2.03i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + 23T^{2} \)
29 \( 1 + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (2.47 + 7.60i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-6.47 + 4.70i)T + (11.4 - 35.1i)T^{2} \)
41 \( 1 + (9.80 + 7.12i)T + (12.6 + 38.9i)T^{2} \)
43 \( 1 - 8.66T + 43T^{2} \)
47 \( 1 + (7.28 + 5.29i)T + (14.5 + 44.6i)T^{2} \)
53 \( 1 + (-1.85 - 5.70i)T + (-42.8 + 31.1i)T^{2} \)
59 \( 1 + (-9.70 + 7.05i)T + (18.2 - 56.1i)T^{2} \)
61 \( 1 + (-2.67 + 8.23i)T + (-49.3 - 35.8i)T^{2} \)
67 \( 1 + 5T + 67T^{2} \)
71 \( 1 + (3.70 - 11.4i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (22.5 - 69.4i)T^{2} \)
79 \( 1 + (3.21 + 9.88i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-1.07 + 3.29i)T + (-67.1 - 48.7i)T^{2} \)
89 \( 1 - 3T + 89T^{2} \)
97 \( 1 + (3.09 + 9.51i)T + (-78.4 + 57.0i)T^{2} \)
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

−10.97208916976209895076170485230, −10.00508099854437345036670365042, −8.909238996689576830370901295892, −8.155955746090247081614470926126, −7.32882632322763125505182539804, −6.41709500351495501172070526838, −5.72860607091586664043349841025, −4.32708753387907450048900259121, −3.74884592172947667493195170078, −1.94604260006466444783753130053, 1.21274042916785195913021637881, 2.50386678133747062772444952217, 3.20430639049972972566745169130, 4.71728455008154047783124927648, 5.26599423968266404275244329761, 6.98560841248599504836349067822, 7.83562932940774176416056750148, 8.655343520608765181955458821880, 9.616022574041415465230181900846, 10.59956727063447584597684720927

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