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Properties

Label 2-546-91.54-c1-0-17
Degree $2$
Conductor $546$
Sign $-0.983 + 0.183i$
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  + (0.707 − 0.707i)2-s + (−0.866 − 0.5i)3-s − 1.00i·4-s + (0.589 − 0.157i)5-s + (−0.965 + 0.258i)6-s + (−1.91 − 1.82i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (0.305 − 0.528i)10-s + (−3.40 + 0.911i)11-s + (−0.500 + 0.866i)12-s + (−1.11 − 3.42i)13-s + (−2.64 + 0.0582i)14-s + (−0.589 − 0.157i)15-s − 1.00·16-s − 0.661·17-s + ⋯
L(s)  = 1  + (0.499 − 0.499i)2-s + (−0.499 − 0.288i)3-s − 0.500i·4-s + (0.263 − 0.0706i)5-s + (−0.394 + 0.105i)6-s + (−0.722 − 0.691i)7-s + (−0.250 − 0.250i)8-s + (0.166 + 0.288i)9-s + (0.0964 − 0.167i)10-s + (−1.02 + 0.274i)11-s + (−0.144 + 0.250i)12-s + (−0.309 − 0.950i)13-s + (−0.706 + 0.0155i)14-s + (−0.152 − 0.0407i)15-s − 0.250·16-s − 0.160·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.983 + 0.183i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.983 + 0.183i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (145, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.983 + 0.183i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0886167 - 0.958046i\)
\(L(\frac12)\) \(\approx\) \(0.0886167 - 0.958046i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.707 + 0.707i)T \)
3 \( 1 + (0.866 + 0.5i)T \)
7 \( 1 + (1.91 + 1.82i)T \)
13 \( 1 + (1.11 + 3.42i)T \)
good5 \( 1 + (-0.589 + 0.157i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.40 - 0.911i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + 0.661T + 17T^{2} \)
19 \( 1 + (-0.476 + 1.77i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + 6.64iT - 23T^{2} \)
29 \( 1 + (-2.81 - 4.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (0.583 - 2.17i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (1.59 + 1.59i)T + 37iT^{2} \)
41 \( 1 + (-1.83 + 6.84i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (6.91 + 3.99i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (1.18 + 4.42i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-5.24 - 9.09i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-6.82 + 6.82i)T - 59iT^{2} \)
61 \( 1 + (-11.3 + 6.57i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (3.70 + 13.8i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-3.44 - 12.8i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.06 + 1.08i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (3.20 - 5.55i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (-5.17 - 5.17i)T + 83iT^{2} \)
89 \( 1 + (-10.6 + 10.6i)T - 89iT^{2} \)
97 \( 1 + (-3.44 + 0.924i)T + (84.0 - 48.5i)T^{2} \)
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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

−10.34714799179047511216303643646, −10.04367464087727279464600103739, −8.678548266782892073333107190959, −7.44526200702919435515887268121, −6.67068870063740637543126245672, −5.56188144779405172179141033952, −4.81842628073723552508854335188, −3.47363291284415174532670772777, −2.30231429200514044371861394046, −0.47199677817147749642917477991, 2.37659085602694697702022398323, 3.64034587183111240296878001919, 4.85633220149247781114855905159, 5.76594823923827076068457663516, 6.37432782105819176714238413485, 7.47397573402478583210995844605, 8.504081892127755847606634449157, 9.611642084214336413907407088530, 10.14320535417970360787370744782, 11.59491625220184064357251956895

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