↑

Properties

Label 2-605-55.9-c1-0-24
Degree $2$
Conductor $605$
Sign $-0.300 + 0.953i$
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  + (−0.492 − 0.159i)2-s + (−1.13 − 1.56i)3-s + (−1.40 − 1.01i)4-s + (1.88 + 1.21i)5-s + (0.309 + 0.951i)6-s + (1.96 − 2.70i)7-s + (1.13 + 1.56i)8-s + (−0.226 + 0.696i)9-s + (−0.732 − 0.896i)10-s + 3.34i·12-s + (4.03 + 1.31i)13-s + (−1.40 + 1.01i)14-s + (−0.243 − 4.31i)15-s + (0.761 + 2.34i)16-s + (3.67 − 1.19i)17-s + (0.222 − 0.306i)18-s + ⋯
L(s)  = 1  + (−0.348 − 0.113i)2-s + (−0.655 − 0.902i)3-s + (−0.700 − 0.509i)4-s + (0.840 + 0.541i)5-s + (0.126 + 0.388i)6-s + (0.743 − 1.02i)7-s + (0.401 + 0.552i)8-s + (−0.0754 + 0.232i)9-s + (−0.231 − 0.283i)10-s + 0.965i·12-s + (1.11 + 0.363i)13-s + (−0.374 + 0.272i)14-s + (−0.0628 − 1.11i)15-s + (0.190 + 0.585i)16-s + (0.891 − 0.289i)17-s + (0.0524 − 0.0722i)18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.300 + 0.953i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(605\)    =    \(5 \cdot 11^{2}\)
Sign: $-0.300 + 0.953i$
Analytic conductor: \(4.83094\)
Root analytic conductor: \(2.19794\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{605} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 605,\ (\ :1/2),\ -0.300 + 0.953i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.649106 - 0.885152i\)
\(L(\frac12)\) \(\approx\) \(0.649106 - 0.885152i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-1.88 - 1.21i)T \)
11 \( 1 \)
good2 \( 1 + (0.492 + 0.159i)T + (1.61 + 1.17i)T^{2} \)
3 \( 1 + (1.13 + 1.56i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-1.96 + 2.70i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (-4.03 - 1.31i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-3.67 + 1.19i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-3.39 + 2.46i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 3.48iT - 23T^{2} \)
29 \( 1 + (5.60 + 4.07i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (2.69 - 8.30i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.05 + 1.45i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (-1.40 + 1.01i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 6.45iT - 43T^{2} \)
47 \( 1 + (6.73 + 9.26i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (2.06 + 0.671i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (1.02 + 0.745i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-3.62 - 11.1i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 2.20iT - 67T^{2} \)
71 \( 1 + (2.53 + 7.79i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (2.87 - 3.96i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.452 + 1.39i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-9.41 + 3.05i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 + 0.464T + 89T^{2} \)
97 \( 1 + (8.69 + 2.82i)T + (78.4 + 57.0i)T^{2} \)
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

−10.52305640621154760839159893584, −9.631615036851048628664332101584, −8.720176875359398488568225827665, −7.56579622487416829618566662973, −6.84327880113255153890330194169, −5.85790942197269640452938298493, −5.09838072133879697263467444102, −3.72416356222382790376451056511, −1.73857491050330333020627052736, −0.902916680014239063248919337429, 1.50487313937443955429907769320, 3.43707996909072736773203083303, 4.58759867720315072613577523517, 5.48510160248403356026771367186, 5.86298326555837699606233288422, 7.78424481048092768447622517096, 8.353637010198966975224190650247, 9.544993485346415648795231018560, 9.588934301166790814154879352702, 10.86347746501017983724032393753

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