Properties

Label 2-546-273.185-c1-0-8
Degree $2$
Conductor $546$
Sign $0.913 + 0.405i$
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  + (−0.866 − 0.5i)2-s + (−1.51 − 0.835i)3-s + (0.499 + 0.866i)4-s + (−0.699 − 1.21i)5-s + (0.895 + 1.48i)6-s + (1.97 + 1.75i)7-s − 0.999i·8-s + (1.60 + 2.53i)9-s + 1.39i·10-s − 1.26i·11-s + (−0.0345 − 1.73i)12-s + (−3.58 − 0.379i)13-s + (−0.831 − 2.51i)14-s + (0.0483 + 2.42i)15-s + (−0.5 + 0.866i)16-s + (3.43 + 5.94i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s + (−0.875 − 0.482i)3-s + (0.249 + 0.433i)4-s + (−0.312 − 0.541i)5-s + (0.365 + 0.605i)6-s + (0.746 + 0.664i)7-s − 0.353i·8-s + (0.534 + 0.845i)9-s + 0.442i·10-s − 0.381i·11-s + (−0.00998 − 0.499i)12-s + (−0.994 − 0.105i)13-s + (−0.222 − 0.671i)14-s + (0.0124 + 0.625i)15-s + (−0.125 + 0.216i)16-s + (0.832 + 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.913 + 0.405i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (185, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.913 + 0.405i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.778708 - 0.165092i\)
\(L(\frac12)\) \(\approx\) \(0.778708 - 0.165092i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + (1.51 + 0.835i)T \)
7 \( 1 + (-1.97 - 1.75i)T \)
13 \( 1 + (3.58 + 0.379i)T \)
good5 \( 1 + (0.699 + 1.21i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 1.26iT - 11T^{2} \)
17 \( 1 + (-3.43 - 5.94i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.61iT - 19T^{2} \)
23 \( 1 + (-2.17 - 1.25i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-4.55 + 2.62i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 + (-4.41 - 2.54i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.45 + 5.98i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.22 + 5.58i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.08 + 10.5i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-1.58 - 2.74i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (-0.444 - 0.256i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-7.18 - 12.4i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 - 3.07iT - 61T^{2} \)
67 \( 1 + 0.624T + 67T^{2} \)
71 \( 1 + (-3.78 - 2.18i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.24 + 4.18i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-7.13 - 12.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 3.52T + 83T^{2} \)
89 \( 1 + (-7.94 + 13.7i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (2.18 + 1.26i)T + (48.5 + 84.0i)T^{2} \)
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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

−10.65347806182433349349652842854, −10.17030532118159234989117017068, −8.797078518131204716718506007898, −8.114451583636029858720021791786, −7.40055109697540875701541988705, −6.06646933313862669049057580509, −5.28553817549843318702204845017, −4.11843089066550491332415922629, −2.31186840781056223455717270338, −1.04040023688907349087146368464, 0.858005646243887381194921201606, 2.92540349958309241837141656148, 4.66147116252141848309112633235, 5.05369027273555247713435996602, 6.62357055645692304335439272279, 7.15539877761195989824911271000, 7.985349216547658831381032495200, 9.414573361412512271869506644507, 9.930003718724476874701956217273, 10.87217289419509749828332849261

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