↑

Properties

Label 2-546-91.80-c1-0-14
Degree $2$
Conductor $546$
Sign $0.692 + 0.721i$
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 + i·3-s + (−0.866 + 0.499i)4-s + (0.739 − 2.75i)5-s + (−0.965 + 0.258i)6-s + (−1.09 − 2.40i)7-s + (−0.707 − 0.707i)8-s − 9-s + 2.85·10-s + (−0.597 − 0.597i)11-s + (−0.499 − 0.866i)12-s + (−1.72 − 3.16i)13-s + (2.04 − 1.68i)14-s + (2.75 + 0.739i)15-s + (0.500 − 0.866i)16-s + (−2.84 − 4.93i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + 0.577i·3-s + (−0.433 + 0.249i)4-s + (0.330 − 1.23i)5-s + (−0.394 + 0.105i)6-s + (−0.414 − 0.909i)7-s + (−0.249 − 0.249i)8-s − 0.333·9-s + 0.902·10-s + (−0.180 − 0.180i)11-s + (−0.144 − 0.249i)12-s + (−0.477 − 0.878i)13-s + (0.545 − 0.449i)14-s + (0.712 + 0.190i)15-s + (0.125 − 0.216i)16-s + (−0.690 − 1.19i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.692 + 0.721i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.692 + 0.721i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (535, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.692 + 0.721i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.07597 - 0.458712i\)
\(L(\frac12)\) \(\approx\) \(1.07597 - 0.458712i\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
7 \( 1 + (1.09 + 2.40i)T \)
13 \( 1 + (1.72 + 3.16i)T \)
good5 \( 1 + (-0.739 + 2.75i)T + (-4.33 - 2.5i)T^{2} \)
11 \( 1 + (0.597 + 0.597i)T + 11iT^{2} \)
17 \( 1 + (2.84 + 4.93i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (1.46 + 1.46i)T + 19iT^{2} \)
23 \( 1 + (-4.10 - 2.37i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.07 + 1.86i)T + (-14.5 + 25.1i)T^{2} \)
31 \( 1 + (1.82 - 0.489i)T + (26.8 - 15.5i)T^{2} \)
37 \( 1 + (-7.19 + 1.92i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (2.29 - 8.55i)T + (-35.5 - 20.5i)T^{2} \)
43 \( 1 + (0.409 + 0.236i)T + (21.5 + 37.2i)T^{2} \)
47 \( 1 + (-11.1 - 2.98i)T + (40.7 + 23.5i)T^{2} \)
53 \( 1 + (3.82 - 6.62i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-6.82 - 1.82i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + 7.89iT - 61T^{2} \)
67 \( 1 + (-2.57 + 2.57i)T - 67iT^{2} \)
71 \( 1 + (-2.65 - 9.90i)T + (-61.4 + 35.5i)T^{2} \)
73 \( 1 + (4.11 + 15.3i)T + (-63.2 + 36.5i)T^{2} \)
79 \( 1 + (5.97 + 10.3i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-1.53 - 1.53i)T + 83iT^{2} \)
89 \( 1 + (2.83 + 10.5i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-10.7 + 2.88i)T + (84.0 - 48.5i)T^{2} \)
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

−10.53387004109021675510115004227, −9.501017029814408868146657360852, −9.094636398410437461963418675638, −7.972475894962086045157897944791, −7.13020436613143310328466741037, −5.90724936710620613929336864235, −4.96929224044216235115462032893, −4.38235010534512648225808995736, −2.99051075749282820889521457385, −0.62427375775732132308029170191, 2.04672253920117952421079499685, 2.67299243431979124449702287838, 3.96791375310878890129701201089, 5.46403615106437963021420757209, 6.42716944177192549143764158120, 7.03662759093996169862731508369, 8.436378548876546950471162215567, 9.246450151201470034395228266933, 10.24120374956339300741150143522, 10.94366093829876635212987949611

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