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Properties

Label 2-546-91.76-c1-0-4
Degree $2$
Conductor $546$
Sign $-0.606 - 0.794i$
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  + (−0.258 + 0.965i)2-s + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (−0.201 + 0.201i)5-s + (−0.258 − 0.965i)6-s + (1.99 + 1.73i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.142 − 0.247i)10-s + (3.76 + 1.00i)11-s + 12-s + (−3.08 + 1.86i)13-s + (−2.19 + 1.48i)14-s + (0.0738 − 0.275i)15-s + (0.500 + 0.866i)16-s + (1.53 − 2.65i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−0.0902 + 0.0902i)5-s + (−0.105 − 0.394i)6-s + (0.755 + 0.655i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.0451 − 0.0781i)10-s + (1.13 + 0.303i)11-s + 0.288·12-s + (−0.856 + 0.516i)13-s + (−0.585 + 0.396i)14-s + (0.0190 − 0.0711i)15-s + (0.125 + 0.216i)16-s + (0.371 − 0.644i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.606 - 0.794i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.606 - 0.794i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.606 - 0.794i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.465933 + 0.941666i\)
\(L(\frac12)\) \(\approx\) \(0.465933 + 0.941666i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.258 - 0.965i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-1.99 - 1.73i)T \)
13 \( 1 + (3.08 - 1.86i)T \)
good5 \( 1 + (0.201 - 0.201i)T - 5iT^{2} \)
11 \( 1 + (-3.76 - 1.00i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-1.53 + 2.65i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.152 - 0.568i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.501 - 0.289i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-5.10 - 8.84i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (6.91 - 6.91i)T - 31iT^{2} \)
37 \( 1 + (7.62 + 2.04i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (7.90 + 2.11i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-0.751 - 0.433i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.05 + 1.05i)T + 47iT^{2} \)
53 \( 1 - 9.83T + 53T^{2} \)
59 \( 1 + (-12.2 + 3.27i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.76 + 1.59i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.37 + 12.5i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.43 - 1.99i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-10.7 - 10.7i)T + 73iT^{2} \)
79 \( 1 + 9.59T + 79T^{2} \)
83 \( 1 + (-10.2 + 10.2i)T - 83iT^{2} \)
89 \( 1 + (3.58 - 13.3i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-2.65 - 9.92i)T + (-84.0 + 48.5i)T^{2} \)
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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

−11.12673046845999251452581730609, −10.11897049126287944077693296335, −9.166368023244243217952223043531, −8.636698326915597891043740292180, −7.22680596829325107075480070723, −6.79696165015338958348773956098, −5.36012012247337630740083233850, −4.94972014949938562599644853201, −3.59127300062637031640291801982, −1.63618256254215479706080799359, 0.74585218020200014024411749999, 2.09576826373360129443899043292, 3.76670070354012114468369927219, 4.62887594005852891975707004928, 5.79294421107900784419021795816, 6.96321643914793213839851529140, 7.915341447560711954641899073154, 8.674918611005755153108304794308, 9.984425048366426324969229350958, 10.42263829792332725823909584917

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