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Properties

Label 2-546-91.88-c1-0-1
Degree $2$
Conductor $546$
Sign $0.0215 - 0.999i$
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  + (−0.866 − 0.5i)2-s − 3-s + (0.499 + 0.866i)4-s + (−0.813 + 0.469i)5-s + (0.866 + 0.5i)6-s + (2.13 + 1.56i)7-s − 0.999i·8-s + 9-s + 0.939·10-s − 1.18i·11-s + (−0.499 − 0.866i)12-s + (−2.78 + 2.28i)13-s + (−1.06 − 2.42i)14-s + (0.813 − 0.469i)15-s + (−0.5 + 0.866i)16-s + (−2.22 − 3.84i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s − 0.577·3-s + (0.249 + 0.433i)4-s + (−0.363 + 0.210i)5-s + (0.353 + 0.204i)6-s + (0.806 + 0.591i)7-s − 0.353i·8-s + 0.333·9-s + 0.297·10-s − 0.357i·11-s + (−0.144 − 0.249i)12-s + (−0.772 + 0.634i)13-s + (−0.284 − 0.647i)14-s + (0.210 − 0.121i)15-s + (−0.125 + 0.216i)16-s + (−0.538 − 0.933i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0215 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0215 - 0.999i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0215 - 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.431979 + 0.422785i\)
\(L(\frac12)\) \(\approx\) \(0.431979 + 0.422785i\)
\(L(\frac{3}{2})\) 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 \( 1 + T \)
7 \( 1 + (-2.13 - 1.56i)T \)
13 \( 1 + (2.78 - 2.28i)T \)
good5 \( 1 + (0.813 - 0.469i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 1.18iT - 11T^{2} \)
17 \( 1 + (2.22 + 3.84i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 2.53iT - 19T^{2} \)
23 \( 1 + (1.31 - 2.27i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-4.86 - 8.41i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.13 + 0.656i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-5.19 - 2.99i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (3.93 - 2.27i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (3.49 - 6.04i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (5.13 - 2.96i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-2.41 + 4.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (5.68 - 3.28i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 - 13.5T + 61T^{2} \)
67 \( 1 - 12.7iT - 67T^{2} \)
71 \( 1 + (6.59 + 3.80i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (7.28 + 4.20i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (5.88 + 10.1i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 12.0iT - 83T^{2} \)
89 \( 1 + (-12.7 - 7.36i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (11.3 + 6.52i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.31413103829017882779322041857, −10.16395115450431402196988016091, −9.330672380458699605626780666427, −8.410683033025432864694706934220, −7.51197050225089351596622915719, −6.65947724231726778387995345412, −5.38935720374465370841256988193, −4.45112667274044804320953813197, −2.96918781440210200071834589772, −1.57877091058893325277929142266, 0.46611362898529670140560049562, 2.13730714193709009237375914592, 4.17595646141431866934872764172, 4.94537819059695580860328637362, 6.11760330013809182586439171448, 7.09706819413801799122915594198, 7.926898082334371436146961344779, 8.581531788529673205185682042615, 9.945786403968972780961025252377, 10.42221400111795272299436565256

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