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Properties

Label 2-2541-231.131-c0-0-3
Degree $2$
Conductor $2541$
Sign $0.915 + 0.402i$
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  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.258 − 0.965i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s − 0.517·13-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s + (−0.258 − 0.965i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.965 − 0.258i)28-s + (−0.866 + 0.5i)31-s + 0.999·36-s + (0.866 − 1.5i)37-s + ⋯
L(s)  = 1  + (0.866 − 0.5i)3-s + (0.5 + 0.866i)4-s + (0.258 − 0.965i)7-s + (0.499 − 0.866i)9-s + (0.866 + 0.499i)12-s − 0.517·13-s + (−0.499 + 0.866i)16-s + (0.707 − 1.22i)19-s + (−0.258 − 0.965i)21-s + (0.5 + 0.866i)25-s − 0.999i·27-s + (0.965 − 0.258i)28-s + (−0.866 + 0.5i)31-s + 0.999·36-s + (0.866 − 1.5i)37-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2541 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.915 + 0.402i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2541\)    =    \(3 \cdot 7 \cdot 11^{2}\)
Sign: $0.915 + 0.402i$
Analytic conductor: \(1.26812\)
Root analytic conductor: \(1.12611\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{2541} (362, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2541,\ (\ :0),\ 0.915 + 0.402i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.873489643\)
\(L(\frac12)\) \(\approx\) \(1.873489643\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 + (-0.866 + 0.5i)T \)
7 \( 1 + (-0.258 + 0.965i)T \)
11 \( 1 \)
good2 \( 1 + (-0.5 - 0.866i)T^{2} \)
5 \( 1 + (-0.5 - 0.866i)T^{2} \)
13 \( 1 + 0.517T + T^{2} \)
17 \( 1 + (0.5 - 0.866i)T^{2} \)
19 \( 1 + (-0.707 + 1.22i)T + (-0.5 - 0.866i)T^{2} \)
23 \( 1 + (0.5 + 0.866i)T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 + (0.866 - 0.5i)T + (0.5 - 0.866i)T^{2} \)
37 \( 1 + (-0.866 + 1.5i)T + (-0.5 - 0.866i)T^{2} \)
41 \( 1 - T^{2} \)
43 \( 1 - 1.93iT - T^{2} \)
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 \( 1 + (0.965 - 1.67i)T + (-0.5 - 0.866i)T^{2} \)
67 \( 1 + (-0.5 + 0.866i)T^{2} \)
71 \( 1 - T^{2} \)
73 \( 1 + (0.965 + 1.67i)T + (-0.5 + 0.866i)T^{2} \)
79 \( 1 + (-1.22 - 0.707i)T + (0.5 + 0.866i)T^{2} \)
83 \( 1 - T^{2} \)
89 \( 1 + (-0.5 - 0.866i)T^{2} \)
97 \( 1 - 1.73iT - T^{2} \)
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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

−9.052082039355900337556340109486, −8.040914860047519648448945575432, −7.40551996100440133922199837017, −7.18322521431494667380325912021, −6.28022862891255239181851735272, −4.90179291085736972153966464803, −4.02171726071065814034793054264, −3.20590809167861397028300858568, −2.48092149985991802975850850141, −1.28625079149349742412569543157, 1.63322496718267075427173345313, 2.40839338756501525207239131531, 3.26332357615987834802157908208, 4.46529178273355965294370317834, 5.25678380701297352071948375718, 5.88680175116002545311669584556, 6.87755051583415233638034076921, 7.76385390001684137396209841061, 8.414471776699171358888831137709, 9.267349724113605788350237677917

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