↑

Properties

Label 2-1155-7.4-c1-0-6
Degree $2$
Conductor $1155$
Sign $0.610 - 0.791i$
Analytic cond. $9.22272$
Root an. cond. $3.03689$
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  + (−1.21 − 2.09i)2-s + (−0.5 + 0.866i)3-s + (−1.93 + 3.35i)4-s + (0.5 + 0.866i)5-s + 2.42·6-s + (−0.185 + 2.63i)7-s + 4.53·8-s + (−0.499 − 0.866i)9-s + (1.21 − 2.09i)10-s + (0.5 − 0.866i)11-s + (−1.93 − 3.35i)12-s + 6.70·13-s + (5.76 − 2.80i)14-s − 0.999·15-s + (−1.62 − 2.80i)16-s + (−2.87 + 4.97i)17-s + ⋯
L(s)  = 1  + (−0.856 − 1.48i)2-s + (−0.288 + 0.499i)3-s + (−0.967 + 1.67i)4-s + (0.223 + 0.387i)5-s + 0.989·6-s + (−0.0701 + 0.997i)7-s + 1.60·8-s + (−0.166 − 0.288i)9-s + (0.383 − 0.663i)10-s + (0.150 − 0.261i)11-s + (−0.558 − 0.967i)12-s + 1.85·13-s + (1.54 − 0.750i)14-s − 0.258·15-s + (−0.405 − 0.701i)16-s + (−0.696 + 1.20i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.610 - 0.791i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $0.610 - 0.791i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (991, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ 0.610 - 0.791i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.6748588241\)
\(L(\frac12)\) \(\approx\) \(0.6748588241\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.5 - 0.866i)T \)
5 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.185 - 2.63i)T \)
11 \( 1 + (-0.5 + 0.866i)T \)
good2 \( 1 + (1.21 + 2.09i)T + (-1 + 1.73i)T^{2} \)
13 \( 1 - 6.70T + 13T^{2} \)
17 \( 1 + (2.87 - 4.97i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 + (3.66 + 6.34i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 6.84T + 29T^{2} \)
31 \( 1 + (0.426 - 0.738i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (-2.08 - 3.61i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + 0.963T + 41T^{2} \)
43 \( 1 - 1.36T + 43T^{2} \)
47 \( 1 + (-6.48 - 11.2i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-4.57 + 7.92i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (6.56 - 11.3i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.28 + 9.16i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (6.57 - 11.3i)T + (-33.5 - 58.0i)T^{2} \)
71 \( 1 + 0.892T + 71T^{2} \)
73 \( 1 + (8.38 - 14.5i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-0.364 - 0.631i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 10.2T + 83T^{2} \)
89 \( 1 + (-1.85 - 3.21i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 - 14.7T + 97T^{2} \)
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

−10.14475620718907645384827536631, −9.100576142680660896398731546450, −8.746041742661272804120567646717, −8.011981483188786564843378580913, −6.27390282458828365253020498392, −5.87534959772257530518557649399, −4.21728903414329751765264784737, −3.52657718991018782780517120204, −2.47966776718869473578898108420, −1.41878720588880606697651541061, 0.44669436113499425057669517162, 1.57198779516221525293504428345, 3.72022893913750200160722876013, 4.89626270480883898227991224938, 5.86615653150373112112380573980, 6.40952314310846230853053873354, 7.48484781507493127176600941646, 7.59077906289879112838850781762, 8.927519646160060806982662905469, 9.184298434277280782012740471954

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