Properties

Label 2-546-91.59-c1-0-3
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: $\chi_{546} (241, \cdot )$
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

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

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