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Properties

Label 2-546-273.254-c1-0-36
Degree $2$
Conductor $546$
Sign $-0.936 - 0.349i$
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 + (−0.701 − 1.58i)3-s + (0.866 + 0.499i)4-s + (−0.996 − 3.71i)5-s + (0.267 + 1.71i)6-s + (0.958 − 2.46i)7-s + (−0.707 − 0.707i)8-s + (−2.01 + 2.22i)9-s + 3.84i·10-s + (−2.84 − 2.84i)11-s + (0.184 − 1.72i)12-s + (3.45 − 1.04i)13-s + (−1.56 + 2.13i)14-s + (−5.19 + 4.18i)15-s + (0.500 + 0.866i)16-s + (2.38 − 4.12i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.404 − 0.914i)3-s + (0.433 + 0.249i)4-s + (−0.445 − 1.66i)5-s + (0.109 + 0.698i)6-s + (0.362 − 0.932i)7-s + (−0.249 − 0.249i)8-s + (−0.672 + 0.740i)9-s + 1.21i·10-s + (−0.857 − 0.857i)11-s + (0.0532 − 0.497i)12-s + (0.957 − 0.288i)13-s + (−0.417 + 0.570i)14-s + (−1.34 + 1.08i)15-s + (0.125 + 0.216i)16-s + (0.578 − 1.00i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.936 - 0.349i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.936 - 0.349i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (527, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.936 - 0.349i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.128187 + 0.710605i\)
\(L(\frac12)\) \(\approx\) \(0.128187 + 0.710605i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.701 + 1.58i)T \)
7 \( 1 + (-0.958 + 2.46i)T \)
13 \( 1 + (-3.45 + 1.04i)T \)
good5 \( 1 + (0.996 + 3.71i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (2.84 + 2.84i)T + 11iT^{2} \)
17 \( 1 + (-2.38 + 4.12i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.21 + 1.21i)T + 19iT^{2} \)
23 \( 1 + (-3.15 - 5.46i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + (-7.51 - 4.33i)T + (14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.191 - 0.714i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (3.35 + 0.900i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (0.359 + 1.34i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (-3.59 + 2.07i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.98 - 1.87i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (4.91 - 2.83i)T + (26.5 - 45.8i)T^{2} \)
59 \( 1 + (4.89 - 1.31i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 - 6.70T + 61T^{2} \)
67 \( 1 + (-6.85 - 6.85i)T + 67iT^{2} \)
71 \( 1 + (9.32 + 2.49i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-9.57 - 2.56i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-1.97 + 3.41i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-7.55 + 7.55i)T - 83iT^{2} \)
89 \( 1 + (2.88 - 10.7i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.13 - 4.24i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.52469768661571349786539612896, −9.174004671322498023951777839281, −8.326570240270402619198889018872, −7.903895251478462783502408609934, −6.97160590339859982411390799993, −5.59043475796284390133126994053, −4.84057985484216211179528200148, −3.26603572032091864694827637920, −1.32073554952484259374208748279, −0.61344779555532770303288803857, 2.37252670747220476890056860685, 3.43503454165622642447985675302, 4.77660718791842947091472972240, 6.09175517501944427711116837759, 6.59132172312272919601983964547, 7.923715340737919134458682461390, 8.560224878557071834953852028818, 9.814336161105938317568380454425, 10.44436587275990918329681213575, 10.98015247079515398086498146988

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