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Properties

Label 2-546-91.19-c1-0-11
Degree $2$
Conductor $546$
Sign $-0.804 + 0.594i$
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 + i·3-s + (0.866 + 0.499i)4-s + (−0.106 + 0.0285i)5-s + (0.258 − 0.965i)6-s + (−1.53 + 2.15i)7-s + (−0.707 − 0.707i)8-s − 9-s + 0.110·10-s + (−3.35 − 3.35i)11-s + (−0.499 + 0.866i)12-s + (−3.58 + 0.403i)13-s + (2.04 − 1.68i)14-s + (−0.0285 − 0.106i)15-s + (0.500 + 0.866i)16-s + (1.45 − 2.51i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (−0.0476 + 0.0127i)5-s + (0.105 − 0.394i)6-s + (−0.580 + 0.814i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.0349·10-s + (−1.01 − 1.01i)11-s + (−0.144 + 0.249i)12-s + (−0.993 + 0.111i)13-s + (0.545 − 0.449i)14-s + (−0.00737 − 0.0275i)15-s + (0.125 + 0.216i)16-s + (0.352 − 0.610i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.804 + 0.594i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.804 + 0.594i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (19, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.804 + 0.594i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0321368 - 0.0976038i\)
\(L(\frac12)\) \(\approx\) \(0.0321368 - 0.0976038i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (1.53 - 2.15i)T \)
13 \( 1 + (3.58 - 0.403i)T \)
good5 \( 1 + (0.106 - 0.0285i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.35 + 3.35i)T + 11iT^{2} \)
17 \( 1 + (-1.45 + 2.51i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.89 + 1.89i)T + 19iT^{2} \)
23 \( 1 + (-4.62 + 2.67i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.975 - 1.68i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-2.51 + 9.39i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.97 - 7.37i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (6.25 - 1.67i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (2.18 - 1.26i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (1.58 + 5.89i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.68 + 6.38i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (0.816 + 3.04i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 7.91iT - 61T^{2} \)
67 \( 1 + (5.73 - 5.73i)T - 67iT^{2} \)
71 \( 1 + (5.73 + 1.53i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (5.15 + 1.38i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (1.92 - 3.32i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-8.03 - 8.03i)T + 83iT^{2} \)
89 \( 1 + (13.6 + 3.66i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (3.07 - 11.4i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.21090193675891734409287046442, −9.671984998561002310866404839885, −8.783708224358634779041261873957, −8.045891670122851804721462329997, −6.91138006452832073577979176482, −5.78315764744497924081922004075, −4.89009276797430279281116265085, −3.24993554822011558377896856432, −2.49302621264752345996075906336, −0.06867135741752078458867884843, 1.76315347178214178628664558679, 3.10398439193897137028460375181, 4.66818635759680361135374822287, 5.85563550454499684131941390183, 7.02271627756790592328350815510, 7.45545823010781878645878041404, 8.304648572778125083690956728069, 9.523868323805337889143544484662, 10.22192280625314669059066700238, 10.82016119758004149712649233639

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