Properties

Label 2-546-91.24-c1-0-12
Degree $2$
Conductor $546$
Sign $0.934 - 0.356i$
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 + i·3-s + (0.866 − 0.499i)4-s + (3.44 + 0.924i)5-s + (0.258 + 0.965i)6-s + (2.64 − 0.0697i)7-s + (0.707 − 0.707i)8-s − 9-s + 3.57·10-s + (−1.42 + 1.42i)11-s + (0.499 + 0.866i)12-s + (−3.59 − 0.320i)13-s + (2.53 − 0.751i)14-s + (−0.924 + 3.44i)15-s + (0.500 − 0.866i)16-s + (−2.36 − 4.09i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (1.54 + 0.413i)5-s + (0.105 + 0.394i)6-s + (0.999 − 0.0263i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s + 1.12·10-s + (−0.428 + 0.428i)11-s + (0.144 + 0.249i)12-s + (−0.996 − 0.0888i)13-s + (0.677 − 0.200i)14-s + (−0.238 + 0.890i)15-s + (0.125 − 0.216i)16-s + (−0.573 − 0.993i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 - 0.356i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.934 - 0.356i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (115, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.934 - 0.356i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.70638 + 0.498809i\)
\(L(\frac12)\) \(\approx\) \(2.70638 + 0.498809i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (-2.64 + 0.0697i)T \)
13 \( 1 + (3.59 + 0.320i)T \)
good5 \( 1 + (-3.44 - 0.924i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.42 - 1.42i)T - 11iT^{2} \)
17 \( 1 + (2.36 + 4.09i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.481 - 0.481i)T - 19iT^{2} \)
23 \( 1 + (6.58 + 3.80i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-3.67 - 6.36i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (2.30 + 8.58i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (-2.36 - 8.84i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-0.562 - 0.150i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.13 + 0.652i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (2.73 - 10.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-6.72 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.23 - 8.32i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 9.88iT - 61T^{2} \)
67 \( 1 + (4.57 + 4.57i)T + 67iT^{2} \)
71 \( 1 + (-5.57 + 1.49i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-3.55 + 0.952i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (2.44 + 4.22i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (7.56 - 7.56i)T - 83iT^{2} \)
89 \( 1 + (14.5 - 3.90i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.35 + 12.5i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.78093509463509591644040090626, −10.01863325573801232977144258306, −9.517369827128952316463294713915, −8.186411924432962533296286683464, −7.01921423490828586665439582252, −6.01186766636088553493260921493, −5.08780090213016577307395749656, −4.50023972771080480059532412417, −2.73355840217821769003465278191, −2.00728580748215264603838737091, 1.71287038367499375553924707222, 2.44992680634636682999875964058, 4.29808327337238597498405831288, 5.43169932396441729910232633166, 5.85670457389948895284967827048, 6.97966021462953099061868120103, 8.027834135045133565473029866396, 8.819814522146588112413322197067, 9.988327671949288038698889109577, 10.77995753069647525442062042063

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