↑

Properties

Label 2-1755-9.4-c1-0-12
Degree $2$
Conductor $1755$
Sign $0.939 + 0.342i$
Analytic cond. $14.0137$
Root an. cond. $3.74349$
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  + (−0.5 − 0.866i)2-s + (0.500 − 0.866i)4-s + (−0.5 + 0.866i)5-s + (0.5 + 0.866i)7-s − 3·8-s + 0.999·10-s + (1 + 1.73i)11-s + (0.5 − 0.866i)13-s + (0.499 − 0.866i)14-s + (0.500 + 0.866i)16-s + 4·17-s + (0.499 + 0.866i)20-s + (0.999 − 1.73i)22-s + (−1.5 + 2.59i)23-s + (−0.499 − 0.866i)25-s − 0.999·26-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s + (0.250 − 0.433i)4-s + (−0.223 + 0.387i)5-s + (0.188 + 0.327i)7-s − 1.06·8-s + 0.316·10-s + (0.301 + 0.522i)11-s + (0.138 − 0.240i)13-s + (0.133 − 0.231i)14-s + (0.125 + 0.216i)16-s + 0.970·17-s + (0.111 + 0.193i)20-s + (0.213 − 0.369i)22-s + (−0.312 + 0.541i)23-s + (−0.0999 − 0.173i)25-s − 0.196·26-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1755 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.939 + 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1755\)    =    \(3^{3} \cdot 5 \cdot 13\)
Sign: $0.939 + 0.342i$
Analytic conductor: \(14.0137\)
Root analytic conductor: \(3.74349\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1755} (1171, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1755,\ (\ :1/2),\ 0.939 + 0.342i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.437968099\)
\(L(\frac12)\) \(\approx\) \(1.437968099\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
5 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (0.5 + 0.866i)T + (-1 + 1.73i)T^{2} \)
7 \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + (-1 - 1.73i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + (1.5 - 2.59i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (4 - 6.92i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 - 4T + 37T^{2} \)
41 \( 1 + (-4.5 + 7.79i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (-4 - 6.92i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-6.5 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 - 10T + 53T^{2} \)
59 \( 1 + (-3 + 5.19i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-0.5 - 0.866i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 - 6T + 71T^{2} \)
73 \( 1 + 12T + 73T^{2} \)
79 \( 1 + (-3 - 5.19i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-5.5 - 9.52i)T + (-41.5 + 71.8i)T^{2} \)
89 \( 1 + 5T + 89T^{2} \)
97 \( 1 + (-1 - 1.73i)T + (-48.5 + 84.0i)T^{2} \)
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

−9.452617172217905317594002533751, −8.669086764310091955242031458881, −7.66959630372023001855795800769, −6.95655666077770984748035838433, −5.93535912207437757333969854450, −5.37191097909780621073518350202, −4.06415913199284873672383073597, −3.07520165112422189546562219507, −2.14353395842943502648732678948, −1.04020999775756195206258519305, 0.75285272130869053092799726812, 2.35619889627858967913008176351, 3.55742716662120237349620349829, 4.24779082216517604527885753431, 5.58466989492990554382649221968, 6.15250757436713236317707307816, 7.24438995068865749478542190592, 7.66730701584450195183143413797, 8.529973070216666159465603107317, 9.037972840979750386523726711932

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