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Properties

Label 2-605-11.9-c1-0-28
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

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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: $\chi_{605} (251, \cdot )$
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 \( 1 + (3.09 - 9.51i)T + (-78.4 - 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

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

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