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Properties

Label 2-546-91.54-c1-0-9
Degree $2$
Conductor $546$
Sign $0.822 + 0.569i$
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  + (−0.707 + 0.707i)2-s + (−0.866 − 0.5i)3-s − 1.00i·4-s + (3.50 − 0.939i)5-s + (0.965 − 0.258i)6-s + (−2.61 − 0.432i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.81 + 3.14i)10-s + (2.71 − 0.726i)11-s + (−0.500 + 0.866i)12-s + (3.16 − 1.71i)13-s + (2.15 − 1.53i)14-s + (−3.50 − 0.939i)15-s − 1.00·16-s − 3.97·17-s + ⋯
L(s)  = 1  + (−0.499 + 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (1.56 − 0.420i)5-s + (0.394 − 0.105i)6-s + (−0.986 − 0.163i)7-s + (0.250 + 0.250i)8-s + (0.166 + 0.288i)9-s + (−0.573 + 0.993i)10-s + (0.817 − 0.219i)11-s + (−0.144 + 0.250i)12-s + (0.879 − 0.476i)13-s + (0.575 − 0.411i)14-s + (−0.905 − 0.242i)15-s − 0.250·16-s − 0.963·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.822 + 0.569i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.822 + 0.569i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.822 + 0.569i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.09446 - 0.341856i\)
\(L(\frac12)\) \(\approx\) \(1.09446 - 0.341856i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 - 0.707i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (2.61 + 0.432i)T \)
13 \( 1 + (-3.16 + 1.71i)T \)
good5 \( 1 + (-3.50 + 0.939i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-2.71 + 0.726i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 3.97T + 17T^{2} \)
19 \( 1 + (0.453 - 1.69i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 2.78iT - 23T^{2} \)
29 \( 1 + (2.41 + 4.19i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-2.48 + 9.26i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (-8.22 - 8.22i)T + 37iT^{2} \)
41 \( 1 + (-2.09 + 7.80i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (1.95 + 1.12i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.538 - 2.00i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-2.21 - 3.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (3.86 - 3.86i)T - 59iT^{2} \)
61 \( 1 + (-11.8 + 6.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-0.946 - 3.53i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (1.90 + 7.12i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.26 - 1.41i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (2.35 - 4.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (4.55 + 4.55i)T + 83iT^{2} \)
89 \( 1 + (-2.30 + 2.30i)T - 89iT^{2} \)
97 \( 1 + (14.9 - 4.00i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.46480109360463397218237952408, −9.698755733837620236507018261213, −9.128885687603681582293595075023, −8.142232074179603409000512241502, −6.71777827846684588468315700586, −6.18044972778905697979325178461, −5.67574169303979646588575116663, −4.20186386515320060020263000070, −2.33072544046236556939982540581, −0.918628830851593893594970285609, 1.50855854210257024487139980559, 2.78629766103736964339185037708, 4.03011678735039262053376725422, 5.49480202052595123779635651807, 6.50233645268230525768178752917, 6.84135586620989034287249522534, 8.774964851570520925279966725756, 9.357824370246725569575513485711, 9.905859566063657213433793570995, 10.81491162875619987476962536261

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