Properties

Label 2-605-11.4-c1-0-21
Degree $2$
Conductor $605$
Sign $-0.394 - 0.918i$
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  + (2.22 + 1.61i)2-s + (−0.0651 + 0.200i)3-s + (1.71 + 5.26i)4-s + (0.809 − 0.587i)5-s + (−0.468 + 0.340i)6-s + (−0.717 − 2.20i)7-s + (−2.99 + 9.22i)8-s + (2.39 + 1.73i)9-s + 2.74·10-s − 1.16·12-s + (0.432 + 0.313i)13-s + (1.97 − 6.06i)14-s + (0.0651 + 0.200i)15-s + (−12.5 + 9.14i)16-s + (−1.95 + 1.42i)17-s + (2.50 + 7.71i)18-s + ⋯
L(s)  = 1  + (1.57 + 1.14i)2-s + (−0.0376 + 0.115i)3-s + (0.855 + 2.63i)4-s + (0.361 − 0.262i)5-s + (−0.191 + 0.138i)6-s + (−0.271 − 0.835i)7-s + (−1.05 + 3.26i)8-s + (0.797 + 0.579i)9-s + 0.867·10-s − 0.336·12-s + (0.119 + 0.0870i)13-s + (0.526 − 1.62i)14-s + (0.0168 + 0.0517i)15-s + (−3.14 + 2.28i)16-s + (−0.475 + 0.345i)17-s + (0.590 + 1.81i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.394 - 0.918i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.394 - 0.918i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.394 - 0.918i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.95742 + 2.97105i\)
\(L(\frac12)\) \(\approx\) \(1.95742 + 2.97105i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.22 - 1.61i)T + (0.618 + 1.90i)T^{2} \)
3 \( 1 + (0.0651 - 0.200i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (0.717 + 2.20i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-0.432 - 0.313i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.95 - 1.42i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.53 + 4.71i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 4.53T + 23T^{2} \)
29 \( 1 + (1.69 + 5.22i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.844 + 0.613i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (-2.31 - 7.12i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-3.27 + 10.0i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 - 4.32T + 43T^{2} \)
47 \( 1 + (-2.09 + 6.43i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.66 - 2.66i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (1.99 + 6.12i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (5.50 - 3.99i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 0.721T + 67T^{2} \)
71 \( 1 + (3.66 - 2.66i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.330 - 1.01i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (3.75 + 2.73i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (11.0 - 7.99i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 12.7T + 89T^{2} \)
97 \( 1 + (3.85 + 2.79i)T + (29.9 + 92.2i)T^{2} \)
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

−11.16441725747129411356532542535, −10.16161481731881343733507824737, −8.897490199277255065379442063196, −7.79008435035543992872819824309, −7.15851566706494808759338016603, −6.33779245545462146215652382476, −5.38733794183417410746364086133, −4.42647510960107961573315789080, −3.86293683441242914741727879587, −2.35662637205342869191387346092, 1.49004957333434367575228644840, 2.58899600883503989750661808529, 3.61012893444894055933446540573, 4.55597146172910740993006088823, 5.78843705640178986683493151933, 6.16100579027448286805301534869, 7.32348053871695115694410573548, 9.169248231077519120668333793018, 9.803159537283974742811210109613, 10.61378109275399896781428007956

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