Properties

Label 2-546-91.59-c1-0-0
Degree $2$
Conductor $546$
Sign $0.230 - 0.973i$
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 + (−4.23 − 1.13i)5-s + (0.965 + 0.258i)6-s + (1.34 − 2.27i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (2.19 + 3.79i)10-s + (3.11 + 0.835i)11-s + (−0.500 − 0.866i)12-s + (−2.90 − 2.13i)13-s + (−2.56 + 0.656i)14-s + (4.23 − 1.13i)15-s − 1.00·16-s − 6.34·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.499i)2-s + (−0.499 + 0.288i)3-s + 0.500i·4-s + (−1.89 − 0.507i)5-s + (0.394 + 0.105i)6-s + (0.509 − 0.860i)7-s + (0.250 − 0.250i)8-s + (0.166 − 0.288i)9-s + (0.692 + 1.19i)10-s + (0.939 + 0.251i)11-s + (−0.144 − 0.250i)12-s + (−0.804 − 0.593i)13-s + (−0.685 + 0.175i)14-s + (1.09 − 0.292i)15-s − 0.250·16-s − 1.53·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.230 - 0.973i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.230 - 0.973i$
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: \((2,\ 546,\ (\ :1/2),\ 0.230 - 0.973i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.233919 + 0.185063i\)
\(L(\frac12)\) \(\approx\) \(0.233919 + 0.185063i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.34 + 2.27i)T \)
13 \( 1 + (2.90 + 2.13i)T \)
good5 \( 1 + (4.23 + 1.13i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-3.11 - 0.835i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + 6.34T + 17T^{2} \)
19 \( 1 + (-1.39 - 5.22i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 - 1.98iT - 23T^{2} \)
29 \( 1 + (1.73 - 3.01i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.21 - 8.25i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-0.127 + 0.127i)T - 37iT^{2} \)
41 \( 1 + (0.00297 + 0.0111i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.69 - 1.55i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.994 + 3.71i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (6.19 - 10.7i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-4.55 - 4.55i)T + 59iT^{2} \)
61 \( 1 + (-4.94 - 2.85i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.37 - 5.13i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-1.61 + 6.00i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.47 + 0.931i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (7.63 + 13.2i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (1.17 - 1.17i)T - 83iT^{2} \)
89 \( 1 + (-12.6 - 12.6i)T + 89iT^{2} \)
97 \( 1 + (-2.25 - 0.603i)T + (84.0 + 48.5i)T^{2} \)
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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

−10.99540001490617160055365219593, −10.39546319061321690879460610259, −9.182477362188751428425201622763, −8.344961446377080224874248028915, −7.51549507918099664504919173998, −6.86002498364691121579483050341, −4.94306691026490199067119306143, −4.23121271767563385578710249233, −3.46758905109915447120742608785, −1.20431867837465364724942342987, 0.25205550367643169393415085619, 2.41745712114822327047684283473, 4.14863007253265748942732585572, 4.90176630773255823338918735985, 6.46744023446792490518905969874, 6.96092240869719442244532079292, 7.906775342765681319043644844650, 8.634384643324623461185509116957, 9.502911470611007380791400375302, 11.04977025648146969836715083917

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