↑

Properties

Label 2-546-91.88-c1-0-0
Degree $2$
Conductor $546$
Sign $0.247 - 0.968i$
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  + (−0.866 − 0.5i)2-s − 3-s + (0.499 + 0.866i)4-s + (−0.620 + 0.358i)5-s + (0.866 + 0.5i)6-s + (−2.52 − 0.792i)7-s − 0.999i·8-s + 9-s + 0.716·10-s − 2.57i·11-s + (−0.499 − 0.866i)12-s + (3.57 − 0.467i)13-s + (1.78 + 1.94i)14-s + (0.620 − 0.358i)15-s + (−0.5 + 0.866i)16-s + (−0.0124 − 0.0215i)17-s + ⋯
L(s)  = 1  + (−0.612 − 0.353i)2-s − 0.577·3-s + (0.249 + 0.433i)4-s + (−0.277 + 0.160i)5-s + (0.353 + 0.204i)6-s + (−0.954 − 0.299i)7-s − 0.353i·8-s + 0.333·9-s + 0.226·10-s − 0.775i·11-s + (−0.144 − 0.249i)12-s + (0.991 − 0.129i)13-s + (0.478 + 0.520i)14-s + (0.160 − 0.0924i)15-s + (−0.125 + 0.216i)16-s + (−0.00301 − 0.00521i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.247 - 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.247 - 0.968i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (361, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.247 - 0.968i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.386623 + 0.300174i\)
\(L(\frac12)\) \(\approx\) \(0.386623 + 0.300174i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.866 + 0.5i)T \)
3 \( 1 + T \)
7 \( 1 + (2.52 + 0.792i)T \)
13 \( 1 + (-3.57 + 0.467i)T \)
good5 \( 1 + (0.620 - 0.358i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + 2.57iT - 11T^{2} \)
17 \( 1 + (0.0124 + 0.0215i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 - 6.91iT - 19T^{2} \)
23 \( 1 + (4.69 - 8.13i)T + (-11.5 - 19.9i)T^{2} \)
29 \( 1 + (2.77 + 4.80i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-4.92 - 2.84i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 + (-3.61 - 2.08i)T + (18.5 + 32.0i)T^{2} \)
41 \( 1 + (-4.54 + 2.62i)T + (20.5 - 35.5i)T^{2} \)
43 \( 1 + (6.37 - 11.0i)T + (-21.5 - 37.2i)T^{2} \)
47 \( 1 + (4.32 - 2.49i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (4.72 - 8.18i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-1.95 + 1.12i)T + (29.5 - 51.0i)T^{2} \)
61 \( 1 + 0.652T + 61T^{2} \)
67 \( 1 + 1.87iT - 67T^{2} \)
71 \( 1 + (-7.52 - 4.34i)T + (35.5 + 61.4i)T^{2} \)
73 \( 1 + (-13.7 - 7.94i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (0.194 + 0.336i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + 8.85iT - 83T^{2} \)
89 \( 1 + (12.8 + 7.41i)T + (44.5 + 77.0i)T^{2} \)
97 \( 1 + (1.78 + 1.03i)T + (48.5 + 84.0i)T^{2} \)
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

−11.09314926418291112524709854376, −9.994504890705755160413314227722, −9.573953975753391942684553718655, −8.233064436222287383574969527789, −7.59823588144733443758218997244, −6.29549603791696208028113090485, −5.81356107340437270916055985109, −3.94750135390532725208965279181, −3.27587723381757939287537078604, −1.33549587749925029921506640631, 0.40136022021364167954458068201, 2.34672975396414332119022067909, 4.00778644319041437879965703138, 5.13613713490860522083369017681, 6.42539998246127625532138680651, 6.69901597357407807373546520424, 8.007058793490598748977627270864, 8.880542824122905877918085512310, 9.723071494778264158196424813907, 10.49433924956102847681451495584

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