↑

Properties

Label 2-1155-5.4-c1-0-9
Degree $2$
Conductor $1155$
Sign $-0.894 + 0.447i$
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  + i·2-s + i·3-s + 4-s + (−2 + i)5-s − 6-s + i·7-s + 3i·8-s − 9-s + (−1 − 2i)10-s − 11-s + i·12-s + 4i·13-s − 14-s + (−1 − 2i)15-s − 16-s + 2i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 0.577i·3-s + 0.5·4-s + (−0.894 + 0.447i)5-s − 0.408·6-s + 0.377i·7-s + 1.06i·8-s − 0.333·9-s + (−0.316 − 0.632i)10-s − 0.301·11-s + 0.288i·12-s + 1.10i·13-s − 0.267·14-s + (−0.258 − 0.516i)15-s − 0.250·16-s + 0.485i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1155 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1155\)    =    \(3 \cdot 5 \cdot 7 \cdot 11\)
Sign: $-0.894 + 0.447i$
Analytic conductor: \(9.22272\)
Root analytic conductor: \(3.03689\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1155} (694, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1155,\ (\ :1/2),\ -0.894 + 0.447i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.125301165\)
\(L(\frac12)\) \(\approx\) \(1.125301165\)
\(L(\frac{3}{2})\) 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 \( 1 + (2 - i)T \)
7 \( 1 - iT \)
11 \( 1 + T \)
good2 \( 1 - iT - 2T^{2} \)
13 \( 1 - 4iT - 13T^{2} \)
17 \( 1 - 2iT - 17T^{2} \)
19 \( 1 + 19T^{2} \)
23 \( 1 + 6iT - 23T^{2} \)
29 \( 1 + 4T + 29T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 - 4iT - 37T^{2} \)
41 \( 1 + 10T + 41T^{2} \)
43 \( 1 + 4iT - 43T^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 - 6iT - 53T^{2} \)
59 \( 1 + 4T + 59T^{2} \)
61 \( 1 + 2T + 61T^{2} \)
67 \( 1 + 8iT - 67T^{2} \)
71 \( 1 + 8T + 71T^{2} \)
73 \( 1 - 73T^{2} \)
79 \( 1 - 14T + 79T^{2} \)
83 \( 1 - 14iT - 83T^{2} \)
89 \( 1 - 12T + 89T^{2} \)
97 \( 1 - 14iT - 97T^{2} \)
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

−10.51113421828337462823454098155, −9.261437518852549106443109776454, −8.428458362809223827938254670442, −7.83466421962417988681777630049, −6.80940568144947170766623991882, −6.32824478092208750211776802413, −5.17844986001079886714265213165, −4.30635701739772528425980030419, −3.21247672631017638303446338122, −2.13665514533546968980186342720, 0.46675791395263709011239999340, 1.64238382775664121014868067820, 3.01302609659908305534976401763, 3.66174311639597372752432973761, 4.94742237210942171762763709176, 5.95001249438461852111411658082, 7.15352748444524550193457558470, 7.55763752519512470532902732689, 8.359359965440661790884563067014, 9.462436573210190299505018527905

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