Properties

Label 2-546-91.54-c1-0-6
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: $\chi_{546} (145, \cdot )$
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

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

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