↑

Properties

Label 2-552-24.11-c1-0-30
Degree $2$
Conductor $552$
Sign $0.503 + 0.863i$
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  + (−0.866 + 1.11i)2-s + (−1.39 − 1.02i)3-s + (−0.499 − 1.93i)4-s − 3.31·5-s + (2.35 − 0.666i)6-s + 4.93i·7-s + (2.59 + 1.11i)8-s + (0.882 + 2.86i)9-s + (2.87 − 3.70i)10-s − 2.29i·11-s + (−1.29 + 3.21i)12-s + 1.77i·13-s + (−5.52 − 4.27i)14-s + (4.62 + 3.41i)15-s + (−3.50 + 1.93i)16-s − 3.10i·17-s + ⋯
L(s)  = 1  + (−0.612 + 0.790i)2-s + (−0.804 − 0.594i)3-s + (−0.249 − 0.968i)4-s − 1.48·5-s + (0.962 − 0.271i)6-s + 1.86i·7-s + (0.918 + 0.395i)8-s + (0.294 + 0.955i)9-s + (0.908 − 1.17i)10-s − 0.693i·11-s + (−0.374 + 0.927i)12-s + 0.492i·13-s + (−1.47 − 1.14i)14-s + (1.19 + 0.881i)15-s + (−0.875 + 0.483i)16-s − 0.753i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 552 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 + 0.863i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(552\)    =    \(2^{3} \cdot 3 \cdot 23\)
Sign: $0.503 + 0.863i$
Analytic conductor: \(4.40774\)
Root analytic conductor: \(2.09946\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{552} (323, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 552,\ (\ :1/2),\ 0.503 + 0.863i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233892 - 0.134390i\)
\(L(\frac12)\) \(\approx\) \(0.233892 - 0.134390i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 - 1.11i)T \)
3 \( 1 + (1.39 + 1.02i)T \)
23 \( 1 + T \)
good5 \( 1 + 3.31T + 5T^{2} \)
7 \( 1 - 4.93iT - 7T^{2} \)
11 \( 1 + 2.29iT - 11T^{2} \)
13 \( 1 - 1.77iT - 13T^{2} \)
17 \( 1 + 3.10iT - 17T^{2} \)
19 \( 1 + 0.712T + 19T^{2} \)
29 \( 1 + 7.87T + 29T^{2} \)
31 \( 1 + 8.99iT - 31T^{2} \)
37 \( 1 + 5.63iT - 37T^{2} \)
41 \( 1 + 4.43iT - 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 - 5.16T + 47T^{2} \)
53 \( 1 + 1.07T + 53T^{2} \)
59 \( 1 - 11.0iT - 59T^{2} \)
61 \( 1 + 2.25iT - 61T^{2} \)
67 \( 1 - 6.72T + 67T^{2} \)
71 \( 1 - 11.8T + 71T^{2} \)
73 \( 1 + 9.31T + 73T^{2} \)
79 \( 1 + 2.14iT - 79T^{2} \)
83 \( 1 + 10.3iT - 83T^{2} \)
89 \( 1 + 4.78iT - 89T^{2} \)
97 \( 1 + 4.06T + 97T^{2} \)
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

−10.97241045134770621242546158306, −9.399581617173478997586741073971, −8.719222282202065409717375199724, −7.82661402344978991389029606968, −7.21683685648074971555650768820, −5.97957774535019986412717529158, −5.51476089496152094878257691017, −4.24833598673582808063985023172, −2.27569871125385375462949322730, −0.27036601160492543572469190270, 1.03090221988579553072377223009, 3.54908820806918672500407968776, 3.99923528648978059052503183380, 4.81215457209024793512687098131, 6.78173839603798233484640416051, 7.48023957874131978410683095731, 8.197455034642041133625958812396, 9.505831802623565945563790774908, 10.38412068660153707550533259068, 10.84491525125763321930729534307

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