↑

Properties

Label 2-546-91.20-c1-0-10
Degree $2$
Conductor $546$
Sign $0.959 + 0.280i$
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 + (−1.14 + 1.14i)5-s + (−0.965 − 0.258i)6-s + (2.54 + 0.717i)7-s + (0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−0.806 + 1.39i)10-s + (−0.325 − 1.21i)11-s − 12-s + (3.51 − 0.808i)13-s + (2.64 + 0.0342i)14-s + (1.55 − 0.417i)15-s + (0.500 − 0.866i)16-s + (−0.338 − 0.586i)17-s + ⋯
L(s)  = 1  + (0.683 − 0.183i)2-s + (−0.499 − 0.288i)3-s + (0.433 − 0.249i)4-s + (−0.510 + 0.510i)5-s + (−0.394 − 0.105i)6-s + (0.962 + 0.271i)7-s + (0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.255 + 0.441i)10-s + (−0.0982 − 0.366i)11-s − 0.288·12-s + (0.974 − 0.224i)13-s + (0.707 + 0.00916i)14-s + (0.402 − 0.107i)15-s + (0.125 − 0.216i)16-s + (−0.0820 − 0.142i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.280i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.959 + 0.280i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (475, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.959 + 0.280i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.94945 - 0.279315i\)
\(L(\frac12)\) \(\approx\) \(1.94945 - 0.279315i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.54 - 0.717i)T \)
13 \( 1 + (-3.51 + 0.808i)T \)
good5 \( 1 + (1.14 - 1.14i)T - 5iT^{2} \)
11 \( 1 + (0.325 + 1.21i)T + (-9.52 + 5.5i)T^{2} \)
17 \( 1 + (0.338 + 0.586i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (-5.48 - 1.46i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + (-2.30 - 1.33i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (-1.92 + 3.32i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-0.505 + 0.505i)T - 31iT^{2} \)
37 \( 1 + (1.16 + 4.34i)T + (-32.0 + 18.5i)T^{2} \)
41 \( 1 + (1.69 + 6.34i)T + (-35.5 + 20.5i)T^{2} \)
43 \( 1 + (1.12 - 0.652i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (-0.433 - 0.433i)T + 47iT^{2} \)
53 \( 1 + 4.52T + 53T^{2} \)
59 \( 1 + (1.06 - 3.97i)T + (-51.0 - 29.5i)T^{2} \)
61 \( 1 + (2.66 - 1.54i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (13.6 - 3.65i)T + (58.0 - 33.5i)T^{2} \)
71 \( 1 + (2.97 - 11.1i)T + (-61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.598 + 0.598i)T + 73iT^{2} \)
79 \( 1 + 9.59T + 79T^{2} \)
83 \( 1 + (0.963 - 0.963i)T - 83iT^{2} \)
89 \( 1 + (8.80 - 2.35i)T + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (-13.9 - 3.73i)T + (84.0 + 48.5i)T^{2} \)
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

−11.13894436348307325275348656559, −10.29080688830877708687131990196, −8.905677293183481791662063614614, −7.82574558075935847317939176328, −7.14413073805724214740737860848, −5.92245465006229443134397586407, −5.27285164552735284778623218031, −4.05855990745180475335249810997, −2.95474710522628957442748864628, −1.36167627171418986703230437776, 1.34756543643392252614741915414, 3.29233285691843002517327270702, 4.52342859163870512937653895243, 4.92807084828109173346264435775, 6.10902626397952392884535299488, 7.16150074305015434667635080551, 8.080007171077161896029093342211, 8.920672850408607494406784788752, 10.21359283371758914000182573571, 11.11066206769469423685398866471

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