↑

Properties

Label 2-546-91.31-c1-0-8
Degree $2$
Conductor $546$
Sign $0.0789 - 0.996i$
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 + (−0.599 + 2.23i)5-s + (0.707 + 0.707i)6-s + (2.03 + 1.69i)7-s + (0.707 + 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.15 + 2.00i)10-s + (−3.26 + 0.875i)11-s + (0.5 + 0.866i)12-s + (−3.59 − 0.240i)13-s + (1.52 + 2.16i)14-s + (−1.63 + 1.63i)15-s + (0.500 + 0.866i)16-s + (1.46 − 2.53i)17-s + ⋯
L(s)  = 1  + (0.683 + 0.183i)2-s + (0.499 + 0.288i)3-s + (0.433 + 0.249i)4-s + (−0.268 + 1.00i)5-s + (0.288 + 0.288i)6-s + (0.768 + 0.640i)7-s + (0.249 + 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.366 + 0.634i)10-s + (−0.984 + 0.263i)11-s + (0.144 + 0.249i)12-s + (−0.997 − 0.0666i)13-s + (0.407 + 0.577i)14-s + (−0.423 + 0.423i)15-s + (0.125 + 0.216i)16-s + (0.354 − 0.614i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0789 - 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0789 - 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.0789 - 0.996i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (31, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.0789 - 0.996i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.78399 + 1.64835i\)
\(L(\frac12)\) \(\approx\) \(1.78399 + 1.64835i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.03 - 1.69i)T \)
13 \( 1 + (3.59 + 0.240i)T \)
good5 \( 1 + (0.599 - 2.23i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (3.26 - 0.875i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-1.46 + 2.53i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-1.68 + 6.28i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (3.91 - 2.26i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 - 5.65T + 29T^{2} \)
31 \( 1 + (-9.59 + 2.57i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-1.15 + 4.30i)T + (-32.0 - 18.5i)T^{2} \)
41 \( 1 + (-1.02 - 1.02i)T + 41iT^{2} \)
43 \( 1 - 9.35iT - 43T^{2} \)
47 \( 1 + (-3.94 - 1.05i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (1.66 - 2.88i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (2.42 + 9.04i)T + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (-4.98 + 2.87i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (2.26 + 8.43i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-0.859 + 0.859i)T - 71iT^{2} \)
73 \( 1 + (3.30 + 12.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (0.0421 + 0.0730i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-3.05 - 3.05i)T + 83iT^{2} \)
89 \( 1 + (6.52 + 1.74i)T + (77.0 + 44.5i)T^{2} \)
97 \( 1 + (1.13 + 1.13i)T + 97iT^{2} \)
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

−11.13866991804169961554341895988, −10.21959355959476582631956993089, −9.322510038202651762728363895946, −7.950390233842422264681016276039, −7.57729405340350668759508650813, −6.47484060609394496779594921333, −5.15782677857527909769671077462, −4.56233537242293322982483516664, −2.91998041513782861326202102672, −2.51161846986287631747549858155, 1.18096846374551267714444836869, 2.58092011359556381693906503241, 4.00264337646178217868171093059, 4.78214546281348327273574093612, 5.71360302774973363746064934682, 7.08555494163469399975830422280, 8.132918923405538745420116413928, 8.363560407094089035075347099473, 10.05709318989575949988521081401, 10.41106809072934505374664600527

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