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Properties

Label 2-546-273.242-c1-0-10
Degree $2$
Conductor $546$
Sign $0.965 + 0.258i$
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 + (−1.11 − 1.32i)3-s + (0.866 + 0.499i)4-s + (1.82 + 0.488i)5-s + (0.737 + 1.56i)6-s + (2.57 − 0.619i)7-s + (−0.707 − 0.707i)8-s + (−0.500 + 2.95i)9-s + (−1.63 − 0.944i)10-s + (1.71 − 0.458i)11-s + (−0.306 − 1.70i)12-s + (1.62 + 3.22i)13-s + (−2.64 − 0.0671i)14-s + (−1.39 − 2.96i)15-s + (0.500 + 0.866i)16-s + (−2.01 + 3.48i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + (−0.645 − 0.763i)3-s + (0.433 + 0.249i)4-s + (0.816 + 0.218i)5-s + (0.301 + 0.639i)6-s + (0.972 − 0.234i)7-s + (−0.249 − 0.249i)8-s + (−0.166 + 0.985i)9-s + (−0.517 − 0.298i)10-s + (0.516 − 0.138i)11-s + (−0.0885 − 0.492i)12-s + (0.449 + 0.893i)13-s + (−0.706 − 0.0179i)14-s + (−0.359 − 0.764i)15-s + (0.125 + 0.216i)16-s + (−0.488 + 0.846i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.965 + 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.965 + 0.258i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (515, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.965 + 0.258i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.10768 - 0.145794i\)
\(L(\frac12)\) \(\approx\) \(1.10768 - 0.145794i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.11 + 1.32i)T \)
7 \( 1 + (-2.57 + 0.619i)T \)
13 \( 1 + (-1.62 - 3.22i)T \)
good5 \( 1 + (-1.82 - 0.488i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-1.71 + 0.458i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (2.01 - 3.48i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.854 - 3.18i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (-3.75 - 6.49i)T + (-11.5 + 19.9i)T^{2} \)
29 \( 1 + 2.36iT - 29T^{2} \)
31 \( 1 + (-7.49 + 2.00i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (7.18 + 1.92i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.16 + 1.16i)T - 41iT^{2} \)
43 \( 1 - 1.15iT - 43T^{2} \)
47 \( 1 + (-2.55 + 9.54i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (-8.17 - 4.71i)T + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-12.1 + 3.26i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (5.96 + 10.3i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (10.6 - 2.84i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (1.54 - 1.54i)T - 71iT^{2} \)
73 \( 1 + (-0.439 - 1.64i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.34 + 5.79i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (4.17 - 4.17i)T - 83iT^{2} \)
89 \( 1 + (0.853 - 3.18i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-5.14 - 5.14i)T + 97iT^{2} \)
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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

−10.79349086515240487553302827653, −10.06604281759755535869190340071, −8.903413371060289336475493884833, −8.135427317799101982520485036455, −7.12555955290196457699833577826, −6.33447887174715090143668360762, −5.48880603193907034786825089062, −4.05111254802229572940450044000, −2.10263360560925768963741143093, −1.38296838966327772551509753991, 1.05622991393188281694069826748, 2.72084777931748800234642079109, 4.52447682223034032979780922069, 5.28307414624516875487444897090, 6.20324343458676980093538345143, 7.14731466718851062606999498873, 8.639578151741584306749442911170, 8.947900434392357254997261869228, 10.05188985738974143689009011963, 10.66729964353895570163675564208

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