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Properties

Label 2-2541-21.11-c0-0-3
Degree $2$
Conductor $2541$
Sign $-0.997 - 0.0633i$
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  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s − 1.73·13-s + (−0.499 − 0.866i)16-s + (−0.866 + 0.499i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (0.866 − 0.499i)28-s + (−0.5 + 0.866i)31-s + 0.999·36-s + (−0.5 − 0.866i)37-s + (−0.866 + 1.49i)39-s + ⋯
L(s)  = 1  + (0.5 − 0.866i)3-s + (−0.5 + 0.866i)4-s + (−0.866 − 0.5i)7-s + (−0.499 − 0.866i)9-s + (0.499 + 0.866i)12-s − 1.73·13-s + (−0.499 − 0.866i)16-s + (−0.866 + 0.499i)21-s + (−0.5 + 0.866i)25-s − 0.999·27-s + (0.866 − 0.499i)28-s + (−0.5 + 0.866i)31-s + 0.999·36-s + (−0.5 − 0.866i)37-s + (−0.866 + 1.49i)39-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0633i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $-0.997 - 0.0633i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (2300, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ -0.997 - 0.0633i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.1944299548\)
\(L(\frac12)\) \(\approx\) \(0.1944299548\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (0.866 + 0.5i)T \)
11 \( 1 \)
good2 \( 1 + (0.5 - 0.866i)T^{2} \)
5 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + 1.73T + T^{2} \)
17 \( 1 + (0.5 + 0.866i)T^{2} \)
19 \( 1 + (-0.5 + 0.866i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 - T^{2} \)
31 \( 1 + (0.5 - 0.866i)T + (-0.5 - 0.866i)T^{2} \)
37 \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 + 1.73T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (0.5 + 0.866i)T^{2} \)
59 \( 1 + (0.5 + 0.866i)T^{2} \)
61 \( 1 + (0.866 + 1.5i)T + (-0.5 + 0.866i)T^{2} \)
67 \( 1 + (-1 + 1.73i)T + (-0.5 - 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.866 - 1.5i)T + (-0.5 - 0.866i)T^{2} \)
79 \( 1 + (-0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (0.5 - 0.866i)T^{2} \)
97 \( 1 + T + T^{2} \)
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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

−8.579608508672570930612890913119, −7.86964327306548030595887580714, −7.09171162499494020801258419024, −6.90906765849760248669704190254, −5.59926636659206104154274711999, −4.63423234785068669534566521831, −3.54212743059437337616221269011, −3.04493592935144394983512087518, −1.95085523612073299686640333303, −0.10978676234213226788710767620, 2.07433171589209253199421299051, 2.89492084903781751331267411186, 3.98569689229862640043562240892, 4.79696788048840205598534849394, 5.41831579134368519716871847577, 6.22537934353792288813758255913, 7.18947985246399559310407866174, 8.203324378031333893256289591515, 8.937746809876331501142072574764, 9.607420252170292440439002810560

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