↑

Properties

Label 2-285-5.4-c1-0-12
Degree $2$
Conductor $285$
Sign $0.997 - 0.0733i$
Analytic cond. $2.27573$
Root an. cond. $1.50855$
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  + 2.47i·2-s − i·3-s − 4.13·4-s + (−0.164 − 2.23i)5-s + 2.47·6-s − 3.36i·7-s − 5.28i·8-s − 9-s + (5.52 − 0.406i)10-s + 3.04·11-s + 4.13i·12-s − 3.21i·13-s + 8.33·14-s + (−2.23 + 0.164i)15-s + 4.81·16-s + 3.46i·17-s + ⋯
L(s)  = 1  + 1.75i·2-s − 0.577i·3-s − 2.06·4-s + (−0.0733 − 0.997i)5-s + 1.01·6-s − 1.27i·7-s − 1.86i·8-s − 0.333·9-s + (1.74 − 0.128i)10-s + 0.916·11-s + 1.19i·12-s − 0.893i·13-s + 2.22·14-s + (−0.575 + 0.0423i)15-s + 1.20·16-s + 0.840i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.997 - 0.0733i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 285 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.997 - 0.0733i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(285\)    =    \(3 \cdot 5 \cdot 19\)
Sign: $0.997 - 0.0733i$
Analytic conductor: \(2.27573\)
Root analytic conductor: \(1.50855\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{285} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 285,\ (\ :1/2),\ 0.997 - 0.0733i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.04904 + 0.0385368i\)
\(L(\frac12)\) \(\approx\) \(1.04904 + 0.0385368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + iT \)
5 \( 1 + (0.164 + 2.23i)T \)
19 \( 1 + T \)
good2 \( 1 - 2.47iT - 2T^{2} \)
7 \( 1 + 3.36iT - 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 + 3.21iT - 13T^{2} \)
17 \( 1 - 3.46iT - 17T^{2} \)
23 \( 1 + 7.11iT - 23T^{2} \)
29 \( 1 - 8.97T + 29T^{2} \)
31 \( 1 + 3.11T + 31T^{2} \)
37 \( 1 - 0.438iT - 37T^{2} \)
41 \( 1 + 2.71T + 41T^{2} \)
43 \( 1 - 7.50iT - 43T^{2} \)
47 \( 1 - 6.31iT - 47T^{2} \)
53 \( 1 + 2.14iT - 53T^{2} \)
59 \( 1 - 11.0T + 59T^{2} \)
61 \( 1 + 7.09T + 61T^{2} \)
67 \( 1 + 13.3iT - 67T^{2} \)
71 \( 1 - 13.3T + 71T^{2} \)
73 \( 1 - 1.62iT - 73T^{2} \)
79 \( 1 - 8.09T + 79T^{2} \)
83 \( 1 - 15.4iT - 83T^{2} \)
89 \( 1 + 5.69T + 89T^{2} \)
97 \( 1 - 8.82iT - 97T^{2} \)
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

−12.34437273355768715481351902639, −10.74267023895193203370389843637, −9.542750859498350037492606883790, −8.369091217144708925585211241831, −8.033035334759471336712142072770, −6.84671525932490409621233701300, −6.19507937991315408243222866849, −4.90801704281784937551365263185, −4.01057215104286459503467276531, −0.854719026914251657156814621388, 2.05357209764584840513181524648, 3.10558076354431072619246257588, 4.08834861489746970175837330626, 5.41128362953880422525437956931, 6.82523152446976273567864333056, 8.638714455483450840334222032028, 9.346140880216993154721453825944, 10.02880329643698755015348928348, 11.07382036405964914706335336194, 11.81025855847775496630002702690

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