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Properties

Label 2-546-91.81-c1-0-4
Degree $2$
Conductor $546$
Sign $0.342 - 0.939i$
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 + (−0.441 − 0.764i)5-s + (−0.5 + 0.866i)6-s + (−2.45 + 0.989i)7-s + 0.999·8-s + 9-s + 0.882·10-s + 1.55·11-s + (−0.499 − 0.866i)12-s + (2.13 + 2.90i)13-s + (0.369 − 2.61i)14-s + (−0.441 − 0.764i)15-s + (−0.5 + 0.866i)16-s + (3.58 + 6.21i)17-s + ⋯
L(s)  = 1  + (−0.353 + 0.612i)2-s + 0.577·3-s + (−0.249 − 0.433i)4-s + (−0.197 − 0.341i)5-s + (−0.204 + 0.353i)6-s + (−0.927 + 0.374i)7-s + 0.353·8-s + 0.333·9-s + 0.279·10-s + 0.467·11-s + (−0.144 − 0.249i)12-s + (0.591 + 0.805i)13-s + (0.0988 − 0.700i)14-s + (−0.113 − 0.197i)15-s + (−0.125 + 0.216i)16-s + (0.870 + 1.50i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.342 - 0.939i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.342 - 0.939i$
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: \((2,\ 546,\ (\ :1/2),\ 0.342 - 0.939i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.08803 + 0.761744i\)
\(L(\frac12)\) \(\approx\) \(1.08803 + 0.761744i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.45 - 0.989i)T \)
13 \( 1 + (-2.13 - 2.90i)T \)
good5 \( 1 + (0.441 + 0.764i)T + (-2.5 + 4.33i)T^{2} \)
11 \( 1 - 1.55T + 11T^{2} \)
17 \( 1 + (-3.58 - 6.21i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 4.74T + 19T^{2} \)
23 \( 1 + (2.64 - 4.58i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (-3.87 - 6.71i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-3.24 + 5.62i)T + (-15.5 - 26.8i)T^{2} \)
37 \( 1 + (0.165 - 0.286i)T + (-18.5 - 32.0i)T^{2} \)
41 \( 1 + (3.02 + 5.24i)T + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.35 + 5.81i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.976 - 1.69i)T + (-23.5 + 40.7i)T^{2} \)
53 \( 1 + (6.74 - 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.63 + 4.56i)T + (-29.5 + 51.0i)T^{2} \)
61 \( 1 + 14.3T + 61T^{2} \)
67 \( 1 + 7.50T + 67T^{2} \)
71 \( 1 + (-5.00 + 8.67i)T + (-35.5 - 61.4i)T^{2} \)
73 \( 1 + (1.93 - 3.34i)T + (-36.5 - 63.2i)T^{2} \)
79 \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 10.5T + 83T^{2} \)
89 \( 1 + (7.40 - 12.8i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 + (-5.79 + 10.0i)T + (-48.5 - 84.0i)T^{2} \)
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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

−10.70644648801024933438752274695, −9.770657590627137874286094436257, −9.111807889804890382947029836519, −8.398983695217684468536687902657, −7.48511709490452308418503177496, −6.44603587588986681531873366374, −5.70234979959816879386657919579, −4.23394200184600898038289000857, −3.26995611477277053822464063235, −1.46013544211365450603518267506, 0.948887359519907235415351781112, 2.95074838898011513045789577194, 3.32366304279400069560725381998, 4.69974784972674400570873042323, 6.21183576884109957262450601453, 7.25536356559560701015541772992, 7.999482028391376619183466328504, 9.070816738850927680269508963789, 9.838289320920193116936931819234, 10.37863694728590001223766709542

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