Properties

Label 2-546-91.33-c1-0-7
Degree $2$
Conductor $546$
Sign $0.734 - 0.678i$
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  + (0.258 − 0.965i)2-s + i·3-s + (−0.866 − 0.499i)4-s + (0.711 + 2.65i)5-s + (0.965 + 0.258i)6-s + (2.33 − 1.23i)7-s + (−0.707 + 0.707i)8-s − 9-s + 2.74·10-s + (−1.56 + 1.56i)11-s + (0.499 − 0.866i)12-s + (0.354 + 3.58i)13-s + (−0.589 − 2.57i)14-s + (−2.65 + 0.711i)15-s + (0.500 + 0.866i)16-s + (−0.838 + 1.45i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + 0.577i·3-s + (−0.433 − 0.249i)4-s + (0.318 + 1.18i)5-s + (0.394 + 0.105i)6-s + (0.883 − 0.467i)7-s + (−0.249 + 0.249i)8-s − 0.333·9-s + 0.868·10-s + (−0.472 + 0.472i)11-s + (0.144 − 0.249i)12-s + (0.0983 + 0.995i)13-s + (−0.157 − 0.689i)14-s + (−0.685 + 0.183i)15-s + (0.125 + 0.216i)16-s + (−0.203 + 0.352i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.734 - 0.678i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.734 - 0.678i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (397, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.734 - 0.678i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.49589 + 0.585368i\)
\(L(\frac12)\) \(\approx\) \(1.49589 + 0.585368i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.258 + 0.965i)T \)
3 \( 1 - iT \)
7 \( 1 + (-2.33 + 1.23i)T \)
13 \( 1 + (-0.354 - 3.58i)T \)
good5 \( 1 + (-0.711 - 2.65i)T + (-4.33 + 2.5i)T^{2} \)
11 \( 1 + (1.56 - 1.56i)T - 11iT^{2} \)
17 \( 1 + (0.838 - 1.45i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-0.129 + 0.129i)T - 19iT^{2} \)
23 \( 1 + (-0.472 + 0.273i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.18 - 3.77i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-7.55 - 2.02i)T + (26.8 + 15.5i)T^{2} \)
37 \( 1 + (1.24 + 0.333i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-1.32 - 4.95i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (2.86 - 1.65i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-9.74 + 2.61i)T + (40.7 - 23.5i)T^{2} \)
53 \( 1 + (-1.78 - 3.08i)T + (-26.5 + 45.8i)T^{2} \)
59 \( 1 + (1.43 - 0.384i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + 13.6iT - 61T^{2} \)
67 \( 1 + (4.98 + 4.98i)T + 67iT^{2} \)
71 \( 1 + (-1.04 + 3.91i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (-2.29 + 8.57i)T + (-63.2 - 36.5i)T^{2} \)
79 \( 1 + (-4.42 + 7.65i)T + (-39.5 - 68.4i)T^{2} \)
83 \( 1 + (11.3 - 11.3i)T - 83iT^{2} \)
89 \( 1 + (-4.16 + 15.5i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (15.6 + 4.18i)T + (84.0 + 48.5i)T^{2} \)
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

−10.81214968020038840549791216905, −10.32787555775686925555472060774, −9.449011320075148860013602091534, −8.391299593154434532796517042754, −7.26026856599896992724696886018, −6.30125939690204729614623896072, −4.99287510912940649702875432796, −4.18919360005673831206261069581, −3.00201902363909505766169648317, −1.85507985139712734344869058567, 0.945757469205167160653286549820, 2.58740766209695261855830620922, 4.34938672922714883167505624335, 5.42659541333913957016003629505, 5.75261660196860482030131878138, 7.19685705159572619943947639183, 8.213497689173619599652696603878, 8.499901118419664670399946543040, 9.519558744649577395587296683530, 10.75563767664644933127112783836

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