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Properties

Label 2-2541-231.227-c0-0-1
Degree $2$
Conductor $2541$
Sign $0.922 + 0.385i$
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 + (−0.838 + 0.544i)7-s + (0.104 − 0.994i)9-s + (0.866 − 0.5i)12-s + (0.418 + 0.304i)13-s + (0.913 + 0.406i)16-s + (1.38 − 0.294i)19-s + (−0.258 + 0.965i)21-s + (−0.669 − 0.743i)25-s + (−0.587 − 0.809i)27-s + (−0.933 + 0.358i)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 + (−0.838 + 0.544i)7-s + (0.104 − 0.994i)9-s + (0.866 − 0.5i)12-s + (0.418 + 0.304i)13-s + (0.913 + 0.406i)16-s + (1.38 − 0.294i)19-s + (−0.258 + 0.965i)21-s + (−0.669 − 0.743i)25-s + (−0.587 − 0.809i)27-s + (−0.933 + 0.358i)28-s + (0.406 + 0.913i)31-s + (0.309 − 0.951i)36-s + (−1.15 + 1.28i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.922 + 0.385i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.922 + 0.385i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (1613, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.922 + 0.385i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.869898251\)
\(L(\frac12)\) \(\approx\) \(1.869898251\)
\(L(1)\) 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 \( 1 + (0.838 - 0.544i)T \)
11 \( 1 \)
good2 \( 1 + (-0.978 - 0.207i)T^{2} \)
5 \( 1 + (0.669 + 0.743i)T^{2} \)
13 \( 1 + (-0.418 - 0.304i)T + (0.309 + 0.951i)T^{2} \)
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 \( 1 + 1.93iT - T^{2} \)
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 \( 1 + (-1.76 - 0.785i)T + (0.669 + 0.743i)T^{2} \)
67 \( 1 + (-0.5 - 0.866i)T^{2} \)
71 \( 1 + (-0.309 + 0.951i)T^{2} \)
73 \( 1 + (1.88 + 0.401i)T + (0.913 + 0.406i)T^{2} \)
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

−8.776950461779287124406298598443, −8.379717783811750123760081015910, −7.30739822553899904291389231455, −6.90679246200424824145656910903, −6.18799497685109794146953699401, −5.39291679026695103408024641757, −3.82604239349986578512826001141, −3.12582369460679287725278182125, −2.43268594559080053861421555563, −1.38324516836234460545431378074, 1.45140770228324604923979676857, 2.69392796235837098168404370247, 3.36284333127398381636376123529, 4.07394538985294847722904547666, 5.35971893718146119559728324540, 5.97467429309078497178058966208, 7.03950802141968160121331601803, 7.55780373657463470829352797063, 8.292101658123763927831597634819, 9.415445415101829032329238099824

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