Properties

Label 2-546-273.152-c1-0-13
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: $\chi_{546} (425, \cdot )$
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

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

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