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Properties

Label 2-546-91.9-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.740 + 0.671i$
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 − 3-s + (−0.499 + 0.866i)4-s + (−1.97 + 3.42i)5-s + (0.5 + 0.866i)6-s + (−1.48 − 2.18i)7-s + 0.999·8-s + 9-s + 3.95·10-s + 4.91·11-s + (0.499 − 0.866i)12-s + (−3.39 + 1.21i)13-s + (−1.15 + 2.38i)14-s + (1.97 − 3.42i)15-s + (−0.5 − 0.866i)16-s + (−0.0702 + 0.121i)17-s + ⋯
L(s)  = 1  + (−0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (−0.883 + 1.52i)5-s + (0.204 + 0.353i)6-s + (−0.561 − 0.827i)7-s + 0.353·8-s + 0.333·9-s + 1.24·10-s + 1.48·11-s + (0.144 − 0.249i)12-s + (−0.941 + 0.338i)13-s + (−0.307 + 0.636i)14-s + (0.509 − 0.883i)15-s + (−0.125 − 0.216i)16-s + (−0.0170 + 0.0295i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.740 + 0.671i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.740 + 0.671i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.740 + 0.671i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.112402 - 0.291254i\)
\(L(\frac12)\) \(\approx\) \(0.112402 - 0.291254i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (1.48 + 2.18i)T \)
13 \( 1 + (3.39 - 1.21i)T \)
good5 \( 1 + (1.97 - 3.42i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 4.91T + 11T^{2} \)
17 \( 1 + (0.0702 - 0.121i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 0.776T + 19T^{2} \)
23 \( 1 + (4.76 + 8.25i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (0.629 - 1.09i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (1.67 + 2.90i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (5.56 + 9.64i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.65 + 8.06i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (0.541 + 0.937i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (-3.33 + 5.77i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (-5.53 - 9.58i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.215 - 0.373i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + 8.28T + 61T^{2} \)
67 \( 1 - 8.19T + 67T^{2} \)
71 \( 1 + (-1.93 - 3.35i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (0.0817 + 0.141i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-2.17 + 3.76i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (-0.536 - 0.929i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (6.54 + 11.3i)T + (-48.5 + 84.0i)T^{2} \)
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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

−10.58712410962681273932638099955, −9.951179541259601058740249506645, −8.875222691457094865911418040711, −7.43830639164429362998335892545, −7.02542556226189722553943763638, −6.19210118848147534363279234735, −4.17979761328932001876527162970, −3.79870671876247846421497330350, −2.37038496840894732881672125353, −0.23412746787435347712020068497, 1.37679228023179005986969849798, 3.73882189180462111423058391629, 4.79903305792873833433470077575, 5.58243580050334378504775166046, 6.56635181872000860938310150546, 7.65895424737127558912763508304, 8.490626445699482059633744226175, 9.352510517202213484517088654193, 9.800413753465427768467347071309, 11.46988341577023304808528489835

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