↑

Properties

Label 2-546-91.4-c1-0-16
Degree $2$
Conductor $546$
Sign $0.793 + 0.608i$
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  + i·2-s + (−0.5 − 0.866i)3-s − 4-s + (3.28 − 1.89i)5-s + (0.866 − 0.5i)6-s + (2.13 − 1.55i)7-s − i·8-s + (−0.499 + 0.866i)9-s + (1.89 + 3.28i)10-s + (0.483 − 0.279i)11-s + (0.5 + 0.866i)12-s + (−2.73 − 2.35i)13-s + (1.55 + 2.13i)14-s + (−3.28 − 1.89i)15-s + 16-s − 6.05·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + (−0.288 − 0.499i)3-s − 0.5·4-s + (1.46 − 0.848i)5-s + (0.353 − 0.204i)6-s + (0.808 − 0.589i)7-s − 0.353i·8-s + (−0.166 + 0.288i)9-s + (0.599 + 1.03i)10-s + (0.145 − 0.0841i)11-s + (0.144 + 0.249i)12-s + (−0.758 − 0.652i)13-s + (0.416 + 0.571i)14-s + (−0.848 − 0.489i)15-s + 0.250·16-s − 1.46·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.793 + 0.608i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $0.793 + 0.608i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (277, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ 0.793 + 0.608i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.50933 - 0.512577i\)
\(L(\frac12)\) \(\approx\) \(1.50933 - 0.512577i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - iT \)
3 \( 1 + (0.5 + 0.866i)T \)
7 \( 1 + (-2.13 + 1.55i)T \)
13 \( 1 + (2.73 + 2.35i)T \)
good5 \( 1 + (-3.28 + 1.89i)T + (2.5 - 4.33i)T^{2} \)
11 \( 1 + (-0.483 + 0.279i)T + (5.5 - 9.52i)T^{2} \)
17 \( 1 + 6.05T + 17T^{2} \)
19 \( 1 + (3.65 + 2.10i)T + (9.5 + 16.4i)T^{2} \)
23 \( 1 - 5.66T + 23T^{2} \)
29 \( 1 + (-1.93 + 3.34i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (-5.96 - 3.44i)T + (15.5 + 26.8i)T^{2} \)
37 \( 1 - 9.24iT - 37T^{2} \)
41 \( 1 + (3.99 + 2.30i)T + (20.5 + 35.5i)T^{2} \)
43 \( 1 + (-6.47 - 11.2i)T + (-21.5 + 37.2i)T^{2} \)
47 \( 1 + (5.18 - 2.99i)T + (23.5 - 40.7i)T^{2} \)
53 \( 1 + (-6.70 + 11.6i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + 2.64iT - 59T^{2} \)
61 \( 1 + (1.78 - 3.09i)T + (-30.5 - 52.8i)T^{2} \)
67 \( 1 + (-10.2 + 5.92i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (4.58 - 2.64i)T + (35.5 - 61.4i)T^{2} \)
73 \( 1 + (2.29 + 1.32i)T + (36.5 + 63.2i)T^{2} \)
79 \( 1 + (-4.38 - 7.60i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 - 6.48iT - 83T^{2} \)
89 \( 1 + 5.45iT - 89T^{2} \)
97 \( 1 + (-4.64 + 2.68i)T + (48.5 - 84.0i)T^{2} \)
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

−10.56855742105943870667637753499, −9.734601291254367358745159262483, −8.743927066414906569390891375781, −8.122080843825482478883704541341, −6.84206641782251815396439648975, −6.27557063695420854754467671683, −4.97297244962891685229379460515, −4.74213082458307926930533297984, −2.38132085443985327350758261061, −1.02104317080090384442113876974, 1.94882872639124819241529399980, 2.61695314302403205358655929333, 4.29177271596715723879542968882, 5.20448015931744344702716220957, 6.15803659383075558741679436383, 7.09330613590821116276411398754, 8.793877947395656729212107482533, 9.208528271498875293421797847069, 10.24555677792367434795978123415, 10.77356696963162974544608574365

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