Properties

Label 2-546-13.3-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.999 + 0.0256i$
Analytic cond. $4.35983$
Root an. cond. $2.08802$
Motivic weight $1$
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)2-s + (0.5 + 0.866i)3-s + (−0.499 + 0.866i)4-s − 2·5-s + (−0.499 + 0.866i)6-s + (−0.5 + 0.866i)7-s − 0.999·8-s + (−0.499 + 0.866i)9-s + (−1 − 1.73i)10-s − 0.999·12-s + (−2.5 + 2.59i)13-s − 0.999·14-s + (−1 − 1.73i)15-s + (−0.5 − 0.866i)16-s + (−0.5 + 0.866i)17-s − 0.999·18-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s + (0.288 + 0.499i)3-s + (−0.249 + 0.433i)4-s − 0.894·5-s + (−0.204 + 0.353i)6-s + (−0.188 + 0.327i)7-s − 0.353·8-s + (−0.166 + 0.288i)9-s + (−0.316 − 0.547i)10-s − 0.288·12-s + (−0.693 + 0.720i)13-s − 0.267·14-s + (−0.258 − 0.447i)15-s + (−0.125 − 0.216i)16-s + (−0.121 + 0.210i)17-s − 0.235·18-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0256i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.999 + 0.0256i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (211, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.999 + 0.0256i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0126870 - 0.989290i\)
\(L(\frac12)\) \(\approx\) \(0.0126870 - 0.989290i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + (-0.5 - 0.866i)T \)
7 \( 1 + (0.5 - 0.866i)T \)
13 \( 1 + (2.5 - 2.59i)T \)
good5 \( 1 + 2T + 5T^{2} \)
11 \( 1 + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (0.5 - 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (3 - 5.19i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (1.5 + 2.59i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (3 + 5.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 5T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-5 - 8.66i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (1.5 - 2.59i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 - 6T + 47T^{2} \)
53 \( 1 - 7T + 53T^{2} \)
59 \( 1 + (-3.5 + 6.06i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (5.5 - 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-6.5 - 11.2i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-1.5 + 2.59i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 12T + 73T^{2} \)
79 \( 1 + 4T + 79T^{2} \)
83 \( 1 + 15T + 83T^{2} \)
89 \( 1 + (-5.5 - 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6 - 10.3i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.44486705536074681205628670098, −10.21058204495526972499839885972, −9.413082034846473580995283774851, −8.332332856124503370979025745578, −7.84165819563095223563449857675, −6.68612452995984429629087174148, −5.73427240241755990394678015777, −4.42219517455802155797093004522, −3.92711560709129080724993777237, −2.50365610393994758614228606872, 0.47488235145224074025443827250, 2.32496429657318601119123205620, 3.43162158435938393683900513849, 4.41117853986702243548973062964, 5.56431147115285942395056643145, 6.90485663741314809921651628365, 7.59384092155122287795381189155, 8.598305082287828719526663275130, 9.524608443729353998170905183427, 10.57698591882422280276716094492

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