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Properties

Label 2-546-273.44-c1-0-11
Degree $2$
Conductor $546$
Sign $0.503 - 0.864i$
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 + (−1.01 + 1.40i)3-s + (0.866 − 0.499i)4-s + (−0.935 + 0.250i)5-s + (−0.617 + 1.61i)6-s + (2.47 + 0.932i)7-s + (0.707 − 0.707i)8-s + (−0.937 − 2.84i)9-s + (−0.839 + 0.484i)10-s + (0.989 + 0.265i)11-s + (−0.178 + 1.72i)12-s + (1.28 + 3.36i)13-s + (2.63 + 0.259i)14-s + (0.598 − 1.56i)15-s + (0.500 − 0.866i)16-s + (2.05 + 3.56i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.586 + 0.810i)3-s + (0.433 − 0.249i)4-s + (−0.418 + 0.112i)5-s + (−0.252 + 0.660i)6-s + (0.935 + 0.352i)7-s + (0.249 − 0.249i)8-s + (−0.312 − 0.949i)9-s + (−0.265 + 0.153i)10-s + (0.298 + 0.0799i)11-s + (−0.0513 + 0.497i)12-s + (0.355 + 0.934i)13-s + (0.703 + 0.0694i)14-s + (0.154 − 0.404i)15-s + (0.125 − 0.216i)16-s + (0.499 + 0.864i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.503 - 0.864i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.503 - 0.864i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (317, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.503 - 0.864i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.58212 + 0.909471i\)
\(L(\frac12)\) \(\approx\) \(1.58212 + 0.909471i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.01 - 1.40i)T \)
7 \( 1 + (-2.47 - 0.932i)T \)
13 \( 1 + (-1.28 - 3.36i)T \)
good5 \( 1 + (0.935 - 0.250i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-0.989 - 0.265i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.05 - 3.56i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-0.715 - 2.67i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (0.318 - 0.551i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 - 3.48iT - 29T^{2} \)
31 \( 1 + (2.41 + 0.646i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.79 + 0.748i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (0.0484 + 0.0484i)T + 41iT^{2} \)
43 \( 1 + 8.67iT - 43T^{2} \)
47 \( 1 + (0.867 + 3.23i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-6.87 + 3.96i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5.60 - 1.50i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (-0.594 + 1.02i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.62 - 0.436i)T + (58.0 + 33.5i)T^{2} \)
71 \( 1 + (8.55 + 8.55i)T + 71iT^{2} \)
73 \( 1 + (-3.42 + 12.7i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (0.0157 - 0.0272i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (6.06 + 6.06i)T + 83iT^{2} \)
89 \( 1 + (1.67 + 6.23i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (8.64 - 8.64i)T - 97iT^{2} \)
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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

−11.07916422361987669861055766491, −10.37636525732653002336580347151, −9.308880765625376982420921522349, −8.362473056811723447656650961336, −7.17707588859280598594981555054, −6.02103591422075232290311546086, −5.29494843061879599903452158650, −4.21281266690111245037565149857, −3.58377794767879819065235917624, −1.72758759658305615856191696541, 1.02725869695746537614565408452, 2.64684710761957838499990091165, 4.15292621898147966541894004937, 5.14178836536868964510404169394, 5.93121292249700740237767593764, 7.07395093210687262765154831195, 7.76189969590396260049234490087, 8.434199601979053777068723662285, 10.01705997797236088659328066748, 11.19044122904380859516516613863

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