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Properties

Label 2-546-91.19-c1-0-7
Degree $2$
Conductor $546$
Sign $0.828 - 0.560i$
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 + (3.74 − 1.00i)5-s + (0.258 − 0.965i)6-s + (−2.20 + 1.46i)7-s + (−0.707 − 0.707i)8-s − 9-s − 3.87·10-s + (3.02 + 3.02i)11-s + (−0.499 + 0.866i)12-s + (3.26 − 1.52i)13-s + (2.50 − 0.842i)14-s + (1.00 + 3.74i)15-s + (0.500 + 0.866i)16-s + (0.792 − 1.37i)17-s + ⋯
L(s)  = 1  + (−0.683 − 0.183i)2-s + 0.577i·3-s + (0.433 + 0.249i)4-s + (1.67 − 0.448i)5-s + (0.105 − 0.394i)6-s + (−0.833 + 0.552i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s − 1.22·10-s + (0.912 + 0.912i)11-s + (−0.144 + 0.249i)12-s + (0.906 − 0.422i)13-s + (0.670 − 0.225i)14-s + (0.258 + 0.965i)15-s + (0.125 + 0.216i)16-s + (0.192 − 0.332i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.828 - 0.560i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.828 - 0.560i$
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: \((2,\ 546,\ (\ :1/2),\ 0.828 - 0.560i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.30444 + 0.399639i\)
\(L(\frac12)\) \(\approx\) \(1.30444 + 0.399639i\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.20 - 1.46i)T \)
13 \( 1 + (-3.26 + 1.52i)T \)
good5 \( 1 + (-3.74 + 1.00i)T + (4.33 - 2.5i)T^{2} \)
11 \( 1 + (-3.02 - 3.02i)T + 11iT^{2} \)
17 \( 1 + (-0.792 + 1.37i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.254 - 0.254i)T + 19iT^{2} \)
23 \( 1 + (6.12 - 3.53i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.17 - 3.75i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.968 + 3.61i)T + (-26.8 - 15.5i)T^{2} \)
37 \( 1 + (0.503 - 1.87i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-5.14 + 1.37i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-7.57 + 4.37i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.515 - 1.92i)T + (-40.7 + 23.5i)T^{2} \)
53 \( 1 + (-4.42 - 7.65i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (-2.31 - 8.63i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 - 5.95iT - 61T^{2} \)
67 \( 1 + (8.80 - 8.80i)T - 67iT^{2} \)
71 \( 1 + (-4.41 - 1.18i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (8.88 + 2.37i)T + (63.2 + 36.5i)T^{2} \)
79 \( 1 + (-8.16 + 14.1i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.65 + 9.65i)T + 83iT^{2} \)
89 \( 1 + (15.8 + 4.23i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (-2.68 + 10.0i)T + (-84.0 - 48.5i)T^{2} \)
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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

−10.39195741561793776819320756950, −9.933081481544470120644845304015, −9.173989382757973530752369395326, −8.840081556137859609272172533515, −7.28841370203332451439874235987, −6.01324992482185432688254221578, −5.72673735794336553856094578687, −4.11359335915345672795773886957, −2.71111899945907890719634228736, −1.52063937768578222345626463112, 1.15087705370901328751023106816, 2.36804056047761592893894605096, 3.71511214340520746780092713133, 5.80480917412817420146015179767, 6.29322367223921518477357899530, 6.79936862122437688025675834535, 8.139879459915223957165603764754, 9.112973982313971342989926037290, 9.720848095004138465782979384741, 10.56934626188806975476788029533

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