Properties

Label 2-546-91.59-c1-0-2
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: $\chi_{546} (241, \cdot )$
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

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

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