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Properties

Label 2-546-13.3-c1-0-8
Degree $2$
Conductor $546$
Sign $0.0128 - 0.999i$
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 + (0.5 + 0.866i)11-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 + ⋯
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 + (0.150 + 0.261i)11-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 + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0128 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0128 - 0.999i$
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: \((2,\ 546,\ (\ :1/2),\ 0.0128 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.54603 + 1.52633i\)
\(L(\frac12)\) \(\approx\) \(1.54603 + 1.52633i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T + (-5.5 + 9.52i)T^{2} \)
17 \( 1 + (-0.5 + 0.866i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.5 - 0.866i)T + (-9.5 - 16.4i)T^{2} \)
23 \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (4.5 + 7.79i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 - 2T + 31T^{2} \)
37 \( 1 + (-1 - 1.73i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (2.5 + 4.33i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-1 + 1.73i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + 7T + 47T^{2} \)
53 \( 1 + T + 53T^{2} \)
59 \( 1 + (5 - 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-5.5 + 9.52i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (1 + 1.73i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (-4 + 6.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 - 8T + 73T^{2} \)
79 \( 1 - 11T + 79T^{2} \)
83 \( 1 + 12T + 83T^{2} \)
89 \( 1 + (5.5 + 9.52i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (-7 + 12.1i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.01634578570366453947207962250, −9.828594102333713680469629293496, −9.373104280125711054227751266293, −8.347068081558194163117139992592, −7.37637192198163924055116237553, −6.33477531013229680853472617202, −5.50439121103564016485082952494, −4.45200911115750619629331998244, −3.47553916883469902126890237556, −1.90369392014556325526361997921, 1.28887603936948580206798661718, 2.47303353765238207728906545702, 3.52371657152434131793503906570, 5.01833616881569129907023582109, 5.90409527694561589222488243218, 6.72438510992968681155549614488, 8.127122846070410330602621664607, 8.884124869224620897261256184805, 9.753253675890985675267943564419, 10.71207294620912451786141981298

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