↑

Properties

Label 2-546-91.76-c1-0-11
Degree $2$
Conductor $546$
Sign $0.998 - 0.0591i$
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 + (−0.866 + 0.5i)3-s + (−0.866 − 0.499i)4-s + (0.301 − 0.301i)5-s + (−0.258 − 0.965i)6-s + (2.59 − 0.536i)7-s + (0.707 − 0.707i)8-s + (0.499 − 0.866i)9-s + (0.213 + 0.369i)10-s + (−5.62 − 1.50i)11-s + 12-s + (2.97 − 2.04i)13-s + (−0.152 + 2.64i)14-s + (−0.110 + 0.412i)15-s + (0.500 + 0.866i)16-s + (3.43 − 5.94i)17-s + ⋯
L(s)  = 1  + (−0.183 + 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (0.134 − 0.134i)5-s + (−0.105 − 0.394i)6-s + (0.979 − 0.202i)7-s + (0.249 − 0.249i)8-s + (0.166 − 0.288i)9-s + (0.0674 + 0.116i)10-s + (−1.69 − 0.454i)11-s + 0.288·12-s + (0.824 − 0.566i)13-s + (−0.0407 + 0.705i)14-s + (−0.0285 + 0.106i)15-s + (0.125 + 0.216i)16-s + (0.832 − 1.44i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0591i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.998 - 0.0591i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (349, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.998 - 0.0591i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.11757 + 0.0330773i\)
\(L(\frac12)\) \(\approx\) \(1.11757 + 0.0330773i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.866 - 0.5i)T \)
7 \( 1 + (-2.59 + 0.536i)T \)
13 \( 1 + (-2.97 + 2.04i)T \)
good5 \( 1 + (-0.301 + 0.301i)T - 5iT^{2} \)
11 \( 1 + (5.62 + 1.50i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.43 + 5.94i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (0.0339 + 0.126i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (-4.00 + 2.31i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (1.99 + 3.45i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.778 + 0.778i)T - 31iT^{2} \)
37 \( 1 + (-3.22 - 0.862i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-10.8 - 2.90i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (-5.32 - 3.07i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-0.709 - 0.709i)T + 47iT^{2} \)
53 \( 1 + 7.17T + 53T^{2} \)
59 \( 1 + (3.84 - 1.02i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (0.320 + 0.184i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.57 + 13.3i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (12.5 - 3.35i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (0.569 + 0.569i)T + 73iT^{2} \)
79 \( 1 - 3.25T + 79T^{2} \)
83 \( 1 + (3.34 - 3.34i)T - 83iT^{2} \)
89 \( 1 + (-1.31 + 4.89i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (3.18 + 11.8i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.92940653357865359265035635895, −9.919491575910025825907781617798, −8.981293192646110186633667163000, −7.86511236931358056056257512429, −7.51236301302693780573686214884, −5.97862908181020799184489986213, −5.31974119425825831737653451066, −4.58821839719488007380285330891, −2.95465998927021933180020073241, −0.844671646644528153945552312127, 1.40507212484560254182602617373, 2.54732045485898436155927553956, 4.10319558623205997712539672998, 5.18313174951441459624878117589, 5.99591913699614589585588806510, 7.47641719001064633488802446382, 8.089698841248321220672523524728, 9.045511346421932158416872175311, 10.35380784564562104391957402152, 10.73322564293610049679119236506

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