Properties

Label 2-552-8.5-c1-0-31
Degree $2$
Conductor $552$
Sign $0.486 + 0.873i$
Analytic cond. $4.40774$
Root an. cond. $2.09946$
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  + (−1.39 + 0.238i)2-s + i·3-s + (1.88 − 0.664i)4-s − 4.40i·5-s + (−0.238 − 1.39i)6-s + 4.93·7-s + (−2.47 + 1.37i)8-s − 9-s + (1.05 + 6.14i)10-s + 0.318i·11-s + (0.664 + 1.88i)12-s − 3.58i·13-s + (−6.88 + 1.17i)14-s + 4.40·15-s + (3.11 − 2.50i)16-s − 2.25·17-s + ⋯
L(s)  = 1  + (−0.985 + 0.168i)2-s + 0.577i·3-s + (0.943 − 0.332i)4-s − 1.97i·5-s + (−0.0973 − 0.569i)6-s + 1.86·7-s + (−0.873 + 0.486i)8-s − 0.333·9-s + (0.332 + 1.94i)10-s + 0.0960i·11-s + (0.191 + 0.544i)12-s − 0.995i·13-s + (−1.83 + 0.314i)14-s + 1.13·15-s + (0.779 − 0.626i)16-s − 0.546·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.486 + 0.873i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.486 + 0.873i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.906057 - 0.532532i\)
\(L(\frac12)\) \(\approx\) \(0.906057 - 0.532532i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.39 - 0.238i)T \)
3 \( 1 - iT \)
23 \( 1 + T \)
good5 \( 1 + 4.40iT - 5T^{2} \)
7 \( 1 - 4.93T + 7T^{2} \)
11 \( 1 - 0.318iT - 11T^{2} \)
13 \( 1 + 3.58iT - 13T^{2} \)
17 \( 1 + 2.25T + 17T^{2} \)
19 \( 1 + 1.36iT - 19T^{2} \)
29 \( 1 - 1.30iT - 29T^{2} \)
31 \( 1 + 4.61T + 31T^{2} \)
37 \( 1 + 7.84iT - 37T^{2} \)
41 \( 1 - 7.38T + 41T^{2} \)
43 \( 1 - 9.90iT - 43T^{2} \)
47 \( 1 - 8.21T + 47T^{2} \)
53 \( 1 + 0.116iT - 53T^{2} \)
59 \( 1 + 10.1iT - 59T^{2} \)
61 \( 1 - 8.74iT - 61T^{2} \)
67 \( 1 + 6.96iT - 67T^{2} \)
71 \( 1 - 3.93T + 71T^{2} \)
73 \( 1 - 0.227T + 73T^{2} \)
79 \( 1 + 1.59T + 79T^{2} \)
83 \( 1 - 3.29iT - 83T^{2} \)
89 \( 1 - 13.8T + 89T^{2} \)
97 \( 1 + 8.63T + 97T^{2} \)
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

−10.65889375819149661380732623534, −9.404570582702183577635297787901, −8.899690762607944931025059942122, −8.112871880538201104754669631597, −7.65335859626603714365534350442, −5.71315306232784713436924787677, −5.13061635132584801727737112420, −4.24790181859662372851824187088, −2.05322533685589677781815972915, −0.869583258696784807257894728226, 1.78260221075951823525107820410, 2.50439217268032452297289499765, 3.97456546194801304299698307372, 5.80908937987921726310053933513, 6.80738624005600038184859894687, 7.40551857849986851489848881028, 8.094426225587254667578372361930, 9.085775041370951365954135973709, 10.34793580933763116919621314803, 10.93776851017061200449054374986

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