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Properties

Label 2-546-91.23-c1-0-8
Degree $2$
Conductor $546$
Sign $0.723 + 0.690i$
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  − i·2-s + (0.5 − 0.866i)3-s − 4-s + (3.80 + 2.19i)5-s + (−0.866 − 0.5i)6-s + (−2.39 + 1.12i)7-s + i·8-s + (−0.499 − 0.866i)9-s + (2.19 − 3.80i)10-s + (0.849 + 0.490i)11-s + (−0.5 + 0.866i)12-s + (0.727 − 3.53i)13-s + (1.12 + 2.39i)14-s + (3.80 − 2.19i)15-s + 16-s + 5.67·17-s + ⋯
L(s)  = 1  − 0.707i·2-s + (0.288 − 0.499i)3-s − 0.5·4-s + (1.70 + 0.982i)5-s + (−0.353 − 0.204i)6-s + (−0.905 + 0.423i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.694 − 1.20i)10-s + (0.256 + 0.147i)11-s + (−0.144 + 0.249i)12-s + (0.201 − 0.979i)13-s + (0.299 + 0.640i)14-s + (0.982 − 0.567i)15-s + 0.250·16-s + 1.37·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.723 + 0.690i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.723 + 0.690i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (205, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.723 + 0.690i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.76246 - 0.706501i\)
\(L(\frac12)\) \(\approx\) \(1.76246 - 0.706501i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + iT \)
3 \( 1 + (-0.5 + 0.866i)T \)
7 \( 1 + (2.39 - 1.12i)T \)
13 \( 1 + (-0.727 + 3.53i)T \)
good5 \( 1 + (-3.80 - 2.19i)T + (2.5 + 4.33i)T^{2} \)
11 \( 1 + (-0.849 - 0.490i)T + (5.5 + 9.52i)T^{2} \)
17 \( 1 - 5.67T + 17T^{2} \)
19 \( 1 + (0.522 - 0.301i)T + (9.5 - 16.4i)T^{2} \)
23 \( 1 - 4.65T + 23T^{2} \)
29 \( 1 + (1.58 + 2.75i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-5.23 + 3.02i)T + (15.5 - 26.8i)T^{2} \)
37 \( 1 + 0.144iT - 37T^{2} \)
41 \( 1 + (8.37 - 4.83i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.17 - 5.50i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (8.79 + 5.07i)T + (23.5 + 40.7i)T^{2} \)
53 \( 1 + (-2.74 - 4.75i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + 8.81iT - 59T^{2} \)
61 \( 1 + (-0.812 - 1.40i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (9.45 + 5.45i)T + (33.5 + 58.0i)T^{2} \)
71 \( 1 + (11.1 + 6.44i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (4.77 - 2.75i)T + (36.5 - 63.2i)T^{2} \)
79 \( 1 + (6.75 - 11.7i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 7.90iT - 83T^{2} \)
89 \( 1 + 1.45iT - 89T^{2} \)
97 \( 1 + (0.165 + 0.0952i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.42177037525956008359996006687, −9.888294009432858270249186384122, −9.338541953313011817765617933426, −8.148833140022281632357708732159, −6.86994814704440806893206488418, −6.09608926106097696283199768313, −5.34296436501615441317379302102, −3.23001007519269439453396702956, −2.79563435271506823255602946122, −1.49098084801636304034477323097, 1.40106047996286010359251346329, 3.17133219439442438591904379255, 4.53061098993202495397928570785, 5.42308635231663131835916661909, 6.23698605124452131967850461663, 7.10616063317760782670427882843, 8.676006796765604960906664522992, 9.017664586626110464933364319322, 9.954847986075712745800527746288, 10.27844924447104304851910922917

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