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Properties

Label 2-2541-231.170-c0-0-1
Degree $2$
Conductor $2541$
Sign $0.955 - 0.294i$
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  + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.406 + 0.913i)7-s + (−0.978 − 0.207i)9-s + (0.499 − 0.866i)12-s + (0.535 + 1.64i)13-s + (0.669 + 0.743i)16-s + (0.866 + 0.499i)21-s + (−0.104 + 0.994i)25-s + (−0.309 + 0.951i)27-s + (−0.743 + 0.669i)28-s + (0.669 − 0.743i)31-s + (−0.809 − 0.587i)36-s + (−0.104 − 0.994i)37-s + (1.69 − 0.360i)39-s + ⋯
L(s)  = 1  + (0.104 − 0.994i)3-s + (0.913 + 0.406i)4-s + (−0.406 + 0.913i)7-s + (−0.978 − 0.207i)9-s + (0.499 − 0.866i)12-s + (0.535 + 1.64i)13-s + (0.669 + 0.743i)16-s + (0.866 + 0.499i)21-s + (−0.104 + 0.994i)25-s + (−0.309 + 0.951i)27-s + (−0.743 + 0.669i)28-s + (0.669 − 0.743i)31-s + (−0.809 − 0.587i)36-s + (−0.104 − 0.994i)37-s + (1.69 − 0.360i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.955 - 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.955 - 0.294i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (632, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.955 - 0.294i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.455497956\)
\(L(\frac12)\) \(\approx\) \(1.455497956\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.104 + 0.994i)T \)
7 \( 1 + (0.406 - 0.913i)T \)
11 \( 1 \)
good2 \( 1 + (-0.913 - 0.406i)T^{2} \)
5 \( 1 + (0.104 - 0.994i)T^{2} \)
13 \( 1 + (-0.535 - 1.64i)T + (-0.809 + 0.587i)T^{2} \)
17 \( 1 + (-0.913 + 0.406i)T^{2} \)
19 \( 1 + (0.669 - 0.743i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + (-0.309 + 0.951i)T^{2} \)
31 \( 1 + (-0.669 + 0.743i)T + (-0.104 - 0.994i)T^{2} \)
37 \( 1 + (0.104 + 0.994i)T + (-0.978 + 0.207i)T^{2} \)
41 \( 1 + (-0.309 - 0.951i)T^{2} \)
43 \( 1 - 1.73T + T^{2} \)
47 \( 1 + (-0.669 + 0.743i)T^{2} \)
53 \( 1 + (0.104 + 0.994i)T^{2} \)
59 \( 1 + (-0.669 - 0.743i)T^{2} \)
61 \( 1 + (1.15 + 1.28i)T + (-0.104 + 0.994i)T^{2} \)
67 \( 1 + (-1 - 1.73i)T + (-0.5 + 0.866i)T^{2} \)
71 \( 1 + (0.809 + 0.587i)T^{2} \)
73 \( 1 + (1.58 + 0.704i)T + (0.669 + 0.743i)T^{2} \)
79 \( 1 + (0.913 + 0.406i)T^{2} \)
83 \( 1 + (0.809 + 0.587i)T^{2} \)
89 \( 1 + (0.5 + 0.866i)T^{2} \)
97 \( 1 + (0.309 + 0.951i)T + (-0.809 + 0.587i)T^{2} \)
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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

−8.978032876840371291755957135288, −8.327146390852252944285278984696, −7.42258594940098202016844071767, −6.89443804049676853476549372830, −6.13318776821818472768997194729, −5.67605774182289659073448508485, −4.16946403636147730900931469360, −3.14641214678061879116583047629, −2.31059418664565113096713007995, −1.59352780617575461617540875036, 0.980924010023325528860118634220, 2.66557876045343184849854766644, 3.25763085885768108066321371347, 4.20815498585879728394966939263, 5.17138401955983251967538725381, 5.97046648151261792476092117278, 6.58298312816194162471140211400, 7.66431296143320302613385105905, 8.176546330238132130494517713213, 9.213147047312922490676236330239

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