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Properties

Label 2-546-91.73-c1-0-7
Degree $2$
Conductor $546$
Sign $0.923 - 0.382i$
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  + (0.258 − 0.965i)2-s + (0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−0.171 − 0.0458i)5-s + (0.707 − 0.707i)6-s + (−1.18 + 2.36i)7-s + (−0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.0885 + 0.153i)10-s + (1.40 + 5.25i)11-s + (−0.5 − 0.866i)12-s + (2.49 + 2.60i)13-s + (1.97 + 1.76i)14-s + (−0.125 − 0.125i)15-s + (0.500 + 0.866i)16-s + (0.452 − 0.784i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−0.0765 − 0.0205i)5-s + (0.288 − 0.288i)6-s + (−0.449 + 0.893i)7-s + (−0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.0280 + 0.0485i)10-s + (0.424 + 1.58i)11-s + (−0.144 − 0.249i)12-s + (0.691 + 0.722i)13-s + (0.527 + 0.470i)14-s + (−0.0323 − 0.0323i)15-s + (0.125 + 0.216i)16-s + (0.109 − 0.190i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 - 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.923 - 0.382i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (73, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.923 - 0.382i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.65248 + 0.328485i\)
\(L(\frac12)\) \(\approx\) \(1.65248 + 0.328485i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 + (-0.866 - 0.5i)T \)
7 \( 1 + (1.18 - 2.36i)T \)
13 \( 1 + (-2.49 - 2.60i)T \)
good5 \( 1 + (0.171 + 0.0458i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.40 - 5.25i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (-0.452 + 0.784i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.207 - 0.0556i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-7.19 + 4.15i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + 0.101T + 29T^{2} \)
31 \( 1 + (-0.239 - 0.893i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (4.21 + 1.12i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-3.40 + 3.40i)T - 41iT^{2} \)
43 \( 1 - 3.97iT - 43T^{2} \)
47 \( 1 + (1.25 - 4.67i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (3.98 - 6.90i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-8.69 + 2.32i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-0.115 + 0.0668i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (8.24 - 2.20i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (-3.61 - 3.61i)T + 71iT^{2} \)
73 \( 1 + (-6.04 + 1.61i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.25 + 12.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-6.96 + 6.96i)T - 83iT^{2} \)
89 \( 1 + (1.75 - 6.53i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-7.96 + 7.96i)T - 97iT^{2} \)
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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

−10.84668201893152713786589088501, −9.844687300013196529618109534935, −9.242741861022189971301461068899, −8.585971967731787522535815496233, −7.25482498673528478625932082579, −6.23689130387250577619955093166, −4.92108018123903093930371948706, −4.09054279843054871110127717816, −2.87633555531259579486231843910, −1.81714887755480356910034768688, 0.954484923661009350149209210192, 3.28517569093518617840635341213, 3.74071694366101929859037105243, 5.35834233790822479943059978685, 6.27594006051647723327025321666, 7.14234978516447481078222920041, 8.026013774550563215561172545828, 8.756417299857801178272340272024, 9.659472851450694082608799528324, 10.77944924404373581675849256850

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