↑

Properties

Label 2-546-91.47-c1-0-11
Degree $2$
Conductor $546$
Sign $0.766 + 0.642i$
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

Related objects

Downloads

Learn more

Normalization:  

Dirichlet series

L(s)  = 1  + (0.965 − 0.258i)2-s + (−0.866 + 0.5i)3-s + (0.866 − 0.499i)4-s + (−0.272 − 1.01i)5-s + (−0.707 + 0.707i)6-s + (2.64 − 0.109i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (−0.526 − 0.912i)10-s + (−1.13 − 0.304i)11-s + (−0.5 + 0.866i)12-s + (0.201 − 3.59i)13-s + (2.52 − 0.789i)14-s + (0.744 + 0.744i)15-s + (0.500 − 0.866i)16-s + (2.53 + 4.39i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.499 + 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.121 − 0.455i)5-s + (−0.288 + 0.288i)6-s + (0.999 − 0.0413i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (−0.166 − 0.288i)10-s + (−0.342 − 0.0916i)11-s + (−0.144 + 0.249i)12-s + (0.0559 − 0.998i)13-s + (0.674 − 0.211i)14-s + (0.192 + 0.192i)15-s + (0.125 − 0.216i)16-s + (0.614 + 1.06i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.766 + 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.766 + 0.642i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (229, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.766 + 0.642i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.87660 - 0.682013i\)
\(L(\frac12)\) \(\approx\) \(1.87660 - 0.682013i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.64 + 0.109i)T \)
13 \( 1 + (-0.201 + 3.59i)T \)
good5 \( 1 + (0.272 + 1.01i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.13 + 0.304i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-2.53 - 4.39i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.54 + 5.78i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-5.58 - 3.22i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 - 3.12T + 29T^{2} \)
31 \( 1 + (3.79 + 1.01i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (0.363 + 1.35i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (-2.05 + 2.05i)T - 41iT^{2} \)
43 \( 1 - 1.78iT - 43T^{2} \)
47 \( 1 + (6.63 - 1.77i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (0.443 + 0.768i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.77 - 6.61i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (9.72 + 5.61i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (3.07 - 11.4i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (-4.62 - 4.62i)T + 71iT^{2} \)
73 \( 1 + (2.31 - 8.65i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (3.14 - 5.45i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (9.80 - 9.80i)T - 83iT^{2} \)
89 \( 1 + (7.08 - 1.89i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (1.59 - 1.59i)T - 97iT^{2} \)
show more
show less
   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−10.94176136272443416382835880678, −10.16971366043943782152633131073, −8.884878868842789896560454166838, −8.018339880324850262548952696525, −6.98070850598109677131990679813, −5.69093741593115791783119182279, −5.08028622273447821568716445337, −4.22940359497067611471067207832, −2.89057533473618094124686117372, −1.16460299210112987132944640203, 1.66954612124831878318572319351, 3.09197474138569197906005928647, 4.52454571652801293006140857517, 5.19813051389547478440746042260, 6.32397724190990454607785469538, 7.18189932514903902000365019431, 7.908178181764174907624368047651, 9.037873526403641249495232563558, 10.39662213825766021355028279821, 11.06931946640034337877516154959

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