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Properties

Label 2-546-91.19-c1-0-13
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: $\chi_{546} (19, \cdot )$
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

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

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