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Properties

Label 2-2541-231.173-c0-0-3
Degree $2$
Conductor $2541$
Sign $0.224 + 0.974i$
Analytic cond. $1.26812$
Root an. cond. $1.12611$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.743 − 0.669i)3-s + (0.978 − 0.207i)4-s + (0.544 − 0.838i)7-s + (0.104 + 0.994i)9-s + (−0.866 − 0.5i)12-s + (1.56 − 1.13i)13-s + (0.913 − 0.406i)16-s + (−1.38 − 0.294i)19-s + (−0.965 + 0.258i)21-s + (−0.669 + 0.743i)25-s + (0.587 − 0.809i)27-s + (0.358 − 0.933i)28-s + (−0.406 + 0.913i)31-s + (0.309 + 0.951i)36-s + (1.15 + 1.28i)37-s + ⋯
L(s)  = 1  + (−0.743 − 0.669i)3-s + (0.978 − 0.207i)4-s + (0.544 − 0.838i)7-s + (0.104 + 0.994i)9-s + (−0.866 − 0.5i)12-s + (1.56 − 1.13i)13-s + (0.913 − 0.406i)16-s + (−1.38 − 0.294i)19-s + (−0.965 + 0.258i)21-s + (−0.669 + 0.743i)25-s + (0.587 − 0.809i)27-s + (0.358 − 0.933i)28-s + (−0.406 + 0.913i)31-s + (0.309 + 0.951i)36-s + (1.15 + 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.224 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.224 + 0.974i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (1328, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.224 + 0.974i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.321941492\)
\(L(\frac12)\) \(\approx\) \(1.321941492\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (0.743 + 0.669i)T \)
7 \( 1 + (-0.544 + 0.838i)T \)
11 \( 1 \)
good2 \( 1 + (-0.978 + 0.207i)T^{2} \)
5 \( 1 + (0.669 - 0.743i)T^{2} \)
13 \( 1 + (-1.56 + 1.13i)T + (0.309 - 0.951i)T^{2} \)
17 \( 1 + (0.978 + 0.207i)T^{2} \)
19 \( 1 + (1.38 + 0.294i)T + (0.913 + 0.406i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.809 - 0.587i)T^{2} \)
31 \( 1 + (0.406 - 0.913i)T + (-0.669 - 0.743i)T^{2} \)
37 \( 1 + (-1.15 - 1.28i)T + (-0.104 + 0.994i)T^{2} \)
41 \( 1 + (0.809 - 0.587i)T^{2} \)
43 \( 1 + 0.517iT - T^{2} \)
47 \( 1 + (0.913 + 0.406i)T^{2} \)
53 \( 1 + (-0.669 - 0.743i)T^{2} \)
59 \( 1 + (0.913 - 0.406i)T^{2} \)
61 \( 1 + (-0.472 + 0.210i)T + (0.669 - 0.743i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (-0.309 - 0.951i)T^{2} \)
73 \( 1 + (0.506 - 0.107i)T + (0.913 - 0.406i)T^{2} \)
79 \( 1 + (1.40 - 0.147i)T + (0.978 - 0.207i)T^{2} \)
83 \( 1 + (-0.309 - 0.951i)T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 + (1.01 + 1.40i)T + (-0.309 + 0.951i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−8.575002942265751811585099041317, −8.019821258204023257858274602587, −7.29766425782025416916741942120, −6.58892335372157789231825882706, −5.97485089677430024183048829984, −5.26278848939379100555376882107, −4.16362214615912426655239332025, −3.07539842727836651911849887775, −1.82288238623627137209669140176, −1.04770002073304835119147729489, 1.58311780912464634107084696336, 2.50106705251299080042289378429, 3.86505705312708923861376872899, 4.31385040980050140519501312347, 5.64413608902341675811188238966, 6.11938236032570468110566419050, 6.62296471278014049601828165744, 7.77711941176291592171860560689, 8.569887399575461739564527865507, 9.186485175796054995753738306130

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