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Properties

Label 2-546-91.41-c1-0-16
Degree $2$
Conductor $546$
Sign $0.225 + 0.974i$
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  + (0.965 + 0.258i)2-s + (−0.866 + 0.5i)3-s + (0.866 + 0.499i)4-s + (−1.10 − 1.10i)5-s + (−0.965 + 0.258i)6-s + (−2.04 − 1.67i)7-s + (0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.778 − 1.34i)10-s + (1.09 − 4.08i)11-s − 12-s + (−2.26 − 2.80i)13-s + (−1.53 − 2.15i)14-s + (1.50 + 0.403i)15-s + (0.500 + 0.866i)16-s + (0.530 − 0.919i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.492 − 0.492i)5-s + (−0.394 + 0.105i)6-s + (−0.772 − 0.634i)7-s + (0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.246 − 0.426i)10-s + (0.330 − 1.23i)11-s − 0.288·12-s + (−0.628 − 0.777i)13-s + (−0.411 − 0.575i)14-s + (0.388 + 0.104i)15-s + (0.125 + 0.216i)16-s + (0.128 − 0.222i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.225 + 0.974i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (223, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.225 + 0.974i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.992796 - 0.789153i\)
\(L(\frac12)\) \(\approx\) \(0.992796 - 0.789153i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.965 - 0.258i)T \)
3 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (2.04 + 1.67i)T \)
13 \( 1 + (2.26 + 2.80i)T \)
good5 \( 1 + (1.10 + 1.10i)T + 5iT^{2} \)
11 \( 1 + (-1.09 + 4.08i)T + (-9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.530 + 0.919i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2.52 + 0.676i)T + (16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.925 - 0.534i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (-2.14 - 3.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.172 + 0.172i)T + 31iT^{2} \)
37 \( 1 + (-1.83 + 6.83i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-0.640 + 2.38i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (9.94 + 5.73i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (6.64 - 6.64i)T - 47iT^{2} \)
53 \( 1 - 3.29T + 53T^{2} \)
59 \( 1 + (-1.79 - 6.70i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-3.62 - 2.09i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-5.87 - 1.57i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.0551 - 0.205i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (-11.8 + 11.8i)T - 73iT^{2} \)
79 \( 1 + 15.8T + 79T^{2} \)
83 \( 1 + (3.21 + 3.21i)T + 83iT^{2} \)
89 \( 1 + (-5.32 - 1.42i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.13 - 0.839i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.72321244495500029537071966553, −9.918382174842380656034027830827, −8.800505225083852644240171169045, −7.76358482336868074308385535146, −6.83198459515202598794496980911, −5.86275972875026202035028431960, −5.00300127053513096174447485705, −3.90406003016015706600034261389, −3.08423225457177372504438440616, −0.61963576882195168893918595508, 1.94203678005128360818652382374, 3.19704527953325348923637186367, 4.38693738078576444356310634074, 5.36131200468206988270103798527, 6.60684145604500608209692214280, 6.93502128374139980267277048507, 8.116976451155606723845410561728, 9.664517349232559275645478222557, 10.00649198694867157541443962085, 11.49147272549531084487839799849

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