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Properties

Label 2-546-91.33-c1-0-1
Degree $2$
Conductor $546$
Sign $-0.854 + 0.520i$
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 + i·3-s + (−0.866 − 0.499i)4-s + (0.939 + 3.50i)5-s + (−0.965 − 0.258i)6-s + (−2.04 − 1.67i)7-s + (0.707 − 0.707i)8-s − 9-s − 3.62·10-s + (−1.98 + 1.98i)11-s + (0.499 − 0.866i)12-s + (−3.16 − 1.71i)13-s + (2.15 − 1.53i)14-s + (−3.50 + 0.939i)15-s + (0.500 + 0.866i)16-s + (−1.98 + 3.44i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.420 + 1.56i)5-s + (−0.394 − 0.105i)6-s + (−0.772 − 0.634i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s − 1.14·10-s + (−0.598 + 0.598i)11-s + (0.144 − 0.249i)12-s + (−0.879 − 0.476i)13-s + (0.575 − 0.411i)14-s + (−0.905 + 0.242i)15-s + (0.125 + 0.216i)16-s + (−0.481 + 0.834i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.854 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.854 + 0.520i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.854 + 0.520i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.189145 - 0.674242i\)
\(L(\frac12)\) \(\approx\) \(0.189145 - 0.674242i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (2.04 + 1.67i)T \)
13 \( 1 + (3.16 + 1.71i)T \)
good5 \( 1 + (-0.939 - 3.50i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.98 - 1.98i)T - 11iT^{2} \)
17 \( 1 + (1.98 - 3.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.23 + 1.23i)T - 19iT^{2} \)
23 \( 1 + (-2.41 + 1.39i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.41 - 4.19i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-9.26 - 2.48i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (11.2 + 3.00i)T + (32.0 + 18.5i)T^{2} \)
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 \( 1 + (-2.00 + 0.538i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-2.21 - 3.83i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (5.27 - 1.41i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 13.6iT - 61T^{2} \)
67 \( 1 + (-2.58 - 2.58i)T + 67iT^{2} \)
71 \( 1 + (1.90 - 7.12i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-1.41 + 5.26i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.35 - 4.08i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-4.55 + 4.55i)T - 83iT^{2} \)
89 \( 1 + (0.842 - 3.14i)T + (-77.0 - 44.5i)T^{2} \)
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

−10.58655644654571748478681213522, −10.51314546229711101388930002732, −9.794536980627226941547222585803, −8.720974528770433907906511975449, −7.33206381966203512936921433407, −6.98261869412467380974707591031, −5.99792064738666914007348960421, −4.89715533436065906776227264613, −3.58538488397551987317943886158, −2.57014117752112451096682688530, 0.40760406778985767887232060971, 1.92968259683440699756416842854, 3.05689267109894013929318629718, 4.72623205049633483690604392504, 5.41907086970311761955289244165, 6.56444070497505885285269809073, 7.915088007906997505135540036564, 8.692619336104919611005596323946, 9.419271110183509497282178794385, 10.00302257387215751225173479426

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