Properties

Label 2-546-91.24-c1-0-6
Degree $2$
Conductor $546$
Sign $0.865 - 0.500i$
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 + (2.82 + 0.756i)5-s + (0.258 + 0.965i)6-s + (−2.36 + 1.17i)7-s + (0.707 − 0.707i)8-s − 9-s + 2.92·10-s + (2.45 − 2.45i)11-s + (0.499 + 0.866i)12-s + (3.36 + 1.28i)13-s + (−1.98 + 1.75i)14-s + (−0.756 + 2.82i)15-s + (0.500 − 0.866i)16-s + (0.623 + 1.07i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 − 0.249i)4-s + (1.26 + 0.338i)5-s + (0.105 + 0.394i)6-s + (−0.895 + 0.445i)7-s + (0.249 − 0.249i)8-s − 0.333·9-s + 0.924·10-s + (0.739 − 0.739i)11-s + (0.144 + 0.249i)12-s + (0.933 + 0.357i)13-s + (−0.529 + 0.468i)14-s + (−0.195 + 0.729i)15-s + (0.125 − 0.216i)16-s + (0.151 + 0.261i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.865 - 0.500i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.865 - 0.500i$
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: \((2,\ 546,\ (\ :1/2),\ 0.865 - 0.500i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.41377 + 0.647004i\)
\(L(\frac12)\) \(\approx\) \(2.41377 + 0.647004i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.36 - 1.17i)T \)
13 \( 1 + (-3.36 - 1.28i)T \)
good5 \( 1 + (-2.82 - 0.756i)T + (4.33 + 2.5i)T^{2} \)
11 \( 1 + (-2.45 + 2.45i)T - 11iT^{2} \)
17 \( 1 + (-0.623 - 1.07i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (3.09 - 3.09i)T - 19iT^{2} \)
23 \( 1 + (1.15 + 0.665i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-0.896 - 1.55i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.443 - 1.65i)T + (-26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.00 + 3.73i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (2.20 + 0.589i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (4.56 + 2.63i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-2.98 + 11.1i)T + (-40.7 - 23.5i)T^{2} \)
53 \( 1 + (2.96 - 5.14i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.57 - 9.61i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + 7.76iT - 61T^{2} \)
67 \( 1 + (2.45 + 2.45i)T + 67iT^{2} \)
71 \( 1 + (9.20 - 2.46i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (-15.5 + 4.15i)T + (63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.83 + 6.64i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (2.31 - 2.31i)T - 83iT^{2} \)
89 \( 1 + (-7.19 + 1.92i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (2.79 + 10.4i)T + (-84.0 + 48.5i)T^{2} \)
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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

−10.69796433164976435601901116008, −10.18163436039498388319284373250, −9.243574721335826213732760627901, −8.553162746585191021825632645361, −6.69768199075128108508867685857, −6.11130760256114212706646953205, −5.52462994645673871356096812576, −3.99737862947852646552187439300, −3.17107227439611556990335150583, −1.86397456283018191194771747945, 1.43253947429979598380912514521, 2.73910250713321341550444650979, 4.05201583609421752317621085395, 5.28085892217441314940721696124, 6.39851604524456816358039108683, 6.57949789859150599229479484993, 7.88331384803777807311307903048, 9.083889814362811649517660407351, 9.787386431245500415667020862424, 10.74732482908756541269106136662

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