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Properties

Label 2-546-91.81-c1-0-0
Degree $2$
Conductor $546$
Sign $-0.295 - 0.955i$
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 + (−0.769 − 1.33i)5-s + (−0.5 + 0.866i)6-s + (−0.131 + 2.64i)7-s − 0.999·8-s + 9-s − 1.53·10-s − 6.38·11-s + (0.499 + 0.866i)12-s + (0.520 + 3.56i)13-s + (2.22 + 1.43i)14-s + (0.769 + 1.33i)15-s + (−0.5 + 0.866i)16-s + (−2.19 − 3.79i)17-s + ⋯
L(s)  = 1  + (0.353 − 0.612i)2-s − 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.344 − 0.596i)5-s + (−0.204 + 0.353i)6-s + (−0.0498 + 0.998i)7-s − 0.353·8-s + 0.333·9-s − 0.486·10-s − 1.92·11-s + (0.144 + 0.249i)12-s + (0.144 + 0.989i)13-s + (0.593 + 0.383i)14-s + (0.198 + 0.344i)15-s + (−0.125 + 0.216i)16-s + (−0.531 − 0.920i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.295 - 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.295 - 0.955i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (445, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.295 - 0.955i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.154005 + 0.208792i\)
\(L(\frac12)\) \(\approx\) \(0.154005 + 0.208792i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.131 - 2.64i)T \)
13 \( 1 + (-0.520 - 3.56i)T \)
good5 \( 1 + (0.769 + 1.33i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 + 6.38T + 11T^{2} \)
17 \( 1 + (2.19 + 3.79i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 0.101T + 19T^{2} \)
23 \( 1 + (4.54 - 7.87i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.51 - 6.08i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.611 - 1.05i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (1.92 - 3.32i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (2.42 + 4.19i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (0.877 - 1.51i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (2.07 + 3.58i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (4.11 - 7.12i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (3.56 + 6.18i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 6.16T + 61T^{2} \)
67 \( 1 + 1.71T + 67T^{2} \)
71 \( 1 + (-4.57 + 7.92i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (-4.82 + 8.35i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (2.03 + 3.52i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 7.02T + 83T^{2} \)
89 \( 1 + (-7.77 + 13.4i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-0.996 + 1.72i)T + (-48.5 - 84.0i)T^{2} \)
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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

−11.23860690333668829575179096918, −10.35517241156575897516070460795, −9.402378714494463742711131315810, −8.573394459584347312048483155303, −7.51903333003267918107349774236, −6.22122348357964659825865157005, −5.16056189362748722398092208147, −4.75968954124652608556195903704, −3.17429964897042730785043956508, −1.92230899134626669703283414713, 0.13482871418623752304333742908, 2.72881980641518619846922454648, 3.99029162104211404013029941581, 4.95714343053320499071200809806, 5.99768207907270280885414343576, 6.83298833095196408484923100350, 7.84469626267041774558131108126, 8.221137691509586835488379191819, 10.07845290789233898120075849516, 10.53548257056924226785344067532

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