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Properties

Label 2-546-91.5-c1-0-5
Degree $2$
Conductor $546$
Sign $0.983 + 0.179i$
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 + (0.698 − 0.187i)5-s + (0.707 + 0.707i)6-s + (2.63 − 0.197i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.361 − 0.626i)10-s + (−0.746 + 2.78i)11-s + (0.5 − 0.866i)12-s + (−3.55 − 0.577i)13-s + (−0.873 − 2.49i)14-s + (−0.511 + 0.511i)15-s + (0.500 − 0.866i)16-s + (2.82 + 4.89i)17-s + ⋯
L(s)  = 1  + (−0.183 − 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 + 0.249i)4-s + (0.312 − 0.0836i)5-s + (0.288 + 0.288i)6-s + (0.997 − 0.0745i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.114 − 0.197i)10-s + (−0.225 + 0.839i)11-s + (0.144 − 0.249i)12-s + (−0.987 − 0.160i)13-s + (−0.233 − 0.667i)14-s + (−0.131 + 0.131i)15-s + (0.125 − 0.216i)16-s + (0.685 + 1.18i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.983 + 0.179i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.983 + 0.179i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (187, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.983 + 0.179i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.23139 - 0.111673i\)
\(L(\frac12)\) \(\approx\) \(1.23139 - 0.111673i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.63 + 0.197i)T \)
13 \( 1 + (3.55 + 0.577i)T \)
good5 \( 1 + (-0.698 + 0.187i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.746 - 2.78i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-2.82 - 4.89i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.58 + 1.49i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (-1.48 - 0.858i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 7.04T + 29T^{2} \)
31 \( 1 + (-1.87 + 6.99i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-8.55 + 2.29i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-8.34 - 8.34i)T + 41iT^{2} \)
43 \( 1 - 1.32iT - 43T^{2} \)
47 \( 1 + (1.67 + 6.23i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-3.51 - 6.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (9.10 + 2.44i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (10.1 + 5.85i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.02 + 0.809i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (5.64 - 5.64i)T - 71iT^{2} \)
73 \( 1 + (-10.5 - 2.83i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (4.87 - 8.44i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (0.644 + 0.644i)T + 83iT^{2} \)
89 \( 1 + (-0.574 - 2.14i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-0.870 - 0.870i)T + 97iT^{2} \)
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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

−10.79631246110174122180019693562, −9.865782864462122116276053044092, −9.480153836033430242290508672593, −8.020329706962749663978431632552, −7.48018432974353067090677449175, −5.92236535095903878263762148803, −4.97669222817394793139043279469, −4.22353729176547128668079931833, −2.64356050255408515649137809878, −1.29465063176257852242949978431, 1.01865242602124806827842645291, 2.78400432214332930497955359159, 4.65358401189921774178286314316, 5.33294247400045082524086130563, 6.19296698259553532342245800056, 7.40846523151268024276197809620, 7.84176219695853945370943842573, 8.989685510688214729253226602544, 9.907464798840992453006036347088, 10.76737383069003694595125189999

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