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Properties

Label 2-546-91.9-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.955 + 0.294i$
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  + (0.5 + 0.866i)2-s − 3-s + (−0.499 + 0.866i)4-s + (−2.07 + 3.58i)5-s + (−0.5 − 0.866i)6-s + (2.11 + 1.59i)7-s − 0.999·8-s + 9-s − 4.14·10-s + 0.523·11-s + (0.499 − 0.866i)12-s + (−3.28 − 1.47i)13-s + (−0.321 + 2.62i)14-s + (2.07 − 3.58i)15-s + (−0.5 − 0.866i)16-s + (1.26 − 2.18i)17-s + ⋯
L(s)  = 1  + (0.353 + 0.612i)2-s − 0.577·3-s + (−0.249 + 0.433i)4-s + (−0.926 + 1.60i)5-s + (−0.204 − 0.353i)6-s + (0.798 + 0.601i)7-s − 0.353·8-s + 0.333·9-s − 1.30·10-s + 0.157·11-s + (0.144 − 0.249i)12-s + (−0.912 − 0.409i)13-s + (−0.0859 + 0.701i)14-s + (0.534 − 0.926i)15-s + (−0.125 − 0.216i)16-s + (0.305 − 0.529i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.955 + 0.294i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.955 + 0.294i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (373, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.955 + 0.294i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.123591 - 0.820941i\)
\(L(\frac12)\) \(\approx\) \(0.123591 - 0.820941i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.5 - 0.866i)T \)
3 \( 1 + T \)
7 \( 1 + (-2.11 - 1.59i)T \)
13 \( 1 + (3.28 + 1.47i)T \)
good5 \( 1 + (2.07 - 3.58i)T + (-2.5 - 4.33i)T^{2} \)
11 \( 1 - 0.523T + 11T^{2} \)
17 \( 1 + (-1.26 + 2.18i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + 5.69T + 19T^{2} \)
23 \( 1 + (-3.69 - 6.40i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.54 - 2.66i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (2.17 + 3.76i)T + (-15.5 + 26.8i)T^{2} \)
37 \( 1 + (-2.83 - 4.90i)T + (-18.5 + 32.0i)T^{2} \)
41 \( 1 + (-2.33 + 4.03i)T + (-20.5 - 35.5i)T^{2} \)
43 \( 1 + (4.81 + 8.34i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.58 - 9.66i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (0.00192 + 0.00332i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (4.05 - 7.02i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 - 6.01T + 61T^{2} \)
67 \( 1 - 3.23T + 67T^{2} \)
71 \( 1 + (-3.98 - 6.90i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 + (-5.99 - 10.3i)T + (-36.5 + 63.2i)T^{2} \)
79 \( 1 + (-1.15 + 2.00i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 - 3.08T + 83T^{2} \)
89 \( 1 + (-5.42 - 9.39i)T + (-44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.31 + 2.28i)T + (-48.5 + 84.0i)T^{2} \)
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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

−11.34803054560201382947794046053, −10.70109465135718458130933476341, −9.580129425840139947613126915602, −8.246225080043922870744325098291, −7.45234084814992842012951241898, −6.88361900167409781853018484493, −5.81861233124943009393143467706, −4.85141382188606782085987890306, −3.70643146179463328657720337125, −2.52623540578326875931186483580, 0.46159957991740011945290513467, 1.77558295864066770436406442467, 3.89889002283052094607472285230, 4.63979861829907439925260688108, 5.09630223569029501515954070673, 6.57443657646155099373415841689, 7.83158189682054647031186661700, 8.533711721435758681792111997520, 9.498329386801067467175906946150, 10.62009208270330813645751147289

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