↑

Properties

Label 2-605-11.3-c1-0-17
Degree $2$
Conductor $605$
Sign $0.999 - 0.0439i$
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  + (0.809 − 0.587i)2-s + (−0.309 + 0.951i)4-s + (−0.809 − 0.587i)5-s + (0.927 + 2.85i)8-s + (2.42 − 1.76i)9-s − 10-s + (1.61 − 1.17i)13-s + (0.809 + 0.587i)16-s + (4.85 + 3.52i)17-s + (0.927 − 2.85i)18-s + (1.23 + 3.80i)19-s + (0.809 − 0.587i)20-s + 4·23-s + (0.309 + 0.951i)25-s + (0.618 − 1.90i)26-s + ⋯
L(s)  = 1  + (0.572 − 0.415i)2-s + (−0.154 + 0.475i)4-s + (−0.361 − 0.262i)5-s + (0.327 + 1.00i)8-s + (0.809 − 0.587i)9-s − 0.316·10-s + (0.448 − 0.326i)13-s + (0.202 + 0.146i)16-s + (1.17 + 0.855i)17-s + (0.218 − 0.672i)18-s + (0.283 + 0.872i)19-s + (0.180 − 0.131i)20-s + 0.834·23-s + (0.0618 + 0.190i)25-s + (0.121 − 0.373i)26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $0.999 - 0.0439i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (366, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ 0.999 - 0.0439i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.97385 + 0.0434035i\)
\(L(\frac12)\) \(\approx\) \(1.97385 + 0.0434035i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (0.809 + 0.587i)T \)
11 \( 1 \)
good2 \( 1 + (-0.809 + 0.587i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (-1.61 + 1.17i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (-4.85 - 3.52i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (-1.23 - 3.80i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 4T + 23T^{2} \)
29 \( 1 + (1.85 - 5.70i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-6.47 + 4.70i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (0.618 - 1.90i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (0.618 + 1.90i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 4T + 43T^{2} \)
47 \( 1 + (3.70 + 11.4i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-1.61 + 1.17i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-1.23 + 3.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (8.09 + 5.87i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 16T + 67T^{2} \)
71 \( 1 + (6.47 + 4.70i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (4.32 - 13.3i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-6.47 + 4.70i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (3.23 + 2.35i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 10T + 89T^{2} \)
97 \( 1 + (8.09 - 5.87i)T + (29.9 - 92.2i)T^{2} \)
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

−10.73756333964133810595275912065, −9.943348825661590613793267692098, −8.801393348590035437164993638353, −8.041356461521959630455450432821, −7.22598422413444394633484261040, −5.93454603405426533661757946872, −4.88160329386011491380724652162, −3.83697140005034931861366967051, −3.23864103470244657020097485634, −1.44255442144260086290961831541, 1.18136033200899559748551070424, 3.02687707869963010462019073037, 4.34140464891098362643813656546, 4.99407969820386545365535348261, 6.07655143256554808635496700798, 7.06187829337472125360715021740, 7.64824315973450778225126623912, 8.987795036755179337317011813612, 9.878108094722218345602200983977, 10.56543984257659172453992149448

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