↑

Properties

Label 2-546-91.76-c1-0-9
Degree $2$
Conductor $546$
Sign $0.257 + 0.966i$
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 + (−2.63 + 2.63i)5-s + (0.258 + 0.965i)6-s + (−1.51 − 2.16i)7-s + (−0.707 + 0.707i)8-s + (0.499 − 0.866i)9-s + (1.86 + 3.22i)10-s + (4.21 + 1.12i)11-s + 12-s + (3.32 − 1.39i)13-s + (−2.48 + 0.903i)14-s + (0.963 − 3.59i)15-s + (0.500 + 0.866i)16-s + (3.14 − 5.44i)17-s + ⋯
L(s)  = 1  + (0.183 − 0.683i)2-s + (−0.499 + 0.288i)3-s + (−0.433 − 0.249i)4-s + (−1.17 + 1.17i)5-s + (0.105 + 0.394i)6-s + (−0.572 − 0.819i)7-s + (−0.249 + 0.249i)8-s + (0.166 − 0.288i)9-s + (0.588 + 1.01i)10-s + (1.27 + 0.340i)11-s + 0.288·12-s + (0.922 − 0.387i)13-s + (−0.664 + 0.241i)14-s + (0.248 − 0.928i)15-s + (0.125 + 0.216i)16-s + (0.762 − 1.32i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.257 + 0.966i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.257 + 0.966i$
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: \((2,\ 546,\ (\ :1/2),\ 0.257 + 0.966i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.753156 - 0.578458i\)
\(L(\frac12)\) \(\approx\) \(0.753156 - 0.578458i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.51 + 2.16i)T \)
13 \( 1 + (-3.32 + 1.39i)T \)
good5 \( 1 + (2.63 - 2.63i)T - 5iT^{2} \)
11 \( 1 + (-4.21 - 1.12i)T + (9.52 + 5.5i)T^{2} \)
17 \( 1 + (-3.14 + 5.44i)T + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (1.27 + 4.76i)T + (-16.4 + 9.5i)T^{2} \)
23 \( 1 + (2.06 - 1.19i)T + (11.5 - 19.9i)T^{2} \)
29 \( 1 + (0.949 + 1.64i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (-0.189 + 0.189i)T - 31iT^{2} \)
37 \( 1 + (-11.2 - 3.01i)T + (32.0 + 18.5i)T^{2} \)
41 \( 1 + (-4.94 - 1.32i)T + (35.5 + 20.5i)T^{2} \)
43 \( 1 + (6.21 + 3.58i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-1.20 - 1.20i)T + 47iT^{2} \)
53 \( 1 - 10.0T + 53T^{2} \)
59 \( 1 + (-9.27 + 2.48i)T + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (10.6 + 6.17i)T + (30.5 + 52.8i)T^{2} \)
67 \( 1 + (-3.16 + 11.8i)T + (-58.0 - 33.5i)T^{2} \)
71 \( 1 + (7.71 - 2.06i)T + (61.4 - 35.5i)T^{2} \)
73 \( 1 + (6.68 + 6.68i)T + 73iT^{2} \)
79 \( 1 + 11.0T + 79T^{2} \)
83 \( 1 + (-4.21 + 4.21i)T - 83iT^{2} \)
89 \( 1 + (1.66 - 6.20i)T + (-77.0 - 44.5i)T^{2} \)
97 \( 1 + (-0.409 - 1.52i)T + (-84.0 + 48.5i)T^{2} \)
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

−10.79962683549546878150271711893, −10.00638268167603032582473939549, −9.186580052891900515247387355479, −7.75109380476531172517453627751, −6.93745177464254969896036797956, −6.15170829399854312012113035225, −4.51647812682213750833591693502, −3.77261840534375560917491166034, −2.99999041049097704598742854719, −0.68611050954939521461693380092, 1.22542168352905879633969639357, 3.73435727041820839557799715978, 4.26000745018103231836451704266, 5.80828771958352231246760110551, 6.10104157905661702722048817139, 7.41008483710006734721240236073, 8.542621943766790547572548550396, 8.662996775224917027064434294374, 9.934351282937442670193661075727, 11.39147455130831797496401670635

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