↑

Properties

Label 2-546-91.6-c1-0-2
Degree $2$
Conductor $546$
Sign $-0.923 - 0.383i$
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 + (2.46 + 2.46i)5-s + (0.258 − 0.965i)6-s + (−1.38 + 2.25i)7-s + (−0.707 − 0.707i)8-s + (0.499 + 0.866i)9-s + (−1.74 + 3.01i)10-s + (−5.73 + 1.53i)11-s + 12-s + (2.00 − 2.99i)13-s + (−2.53 − 0.755i)14-s + (−0.901 − 3.36i)15-s + (0.500 − 0.866i)16-s + (0.681 + 1.18i)17-s + ⋯
L(s)  = 1  + (0.183 + 0.683i)2-s + (−0.499 − 0.288i)3-s + (−0.433 + 0.249i)4-s + (1.10 + 1.10i)5-s + (0.105 − 0.394i)6-s + (−0.523 + 0.851i)7-s + (−0.249 − 0.249i)8-s + (0.166 + 0.288i)9-s + (−0.550 + 0.953i)10-s + (−1.72 + 0.463i)11-s + 0.288·12-s + (0.555 − 0.831i)13-s + (−0.677 − 0.201i)14-s + (−0.232 − 0.868i)15-s + (0.125 − 0.216i)16-s + (0.165 + 0.286i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 546 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.923 - 0.383i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(546\)    =    \(2 \cdot 3 \cdot 7 \cdot 13\)
Sign: $-0.923 - 0.383i$
Analytic conductor: \(4.35983\)
Root analytic conductor: \(2.08802\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{546} (97, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 546,\ (\ :1/2),\ -0.923 - 0.383i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.210386 + 1.05656i\)
\(L(\frac12)\) \(\approx\) \(0.210386 + 1.05656i\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.38 - 2.25i)T \)
13 \( 1 + (-2.00 + 2.99i)T \)
good5 \( 1 + (-2.46 - 2.46i)T + 5iT^{2} \)
11 \( 1 + (5.73 - 1.53i)T + (9.52 - 5.5i)T^{2} \)
17 \( 1 + (-0.681 - 1.18i)T + (-8.5 + 14.7i)T^{2} \)
19 \( 1 + (0.203 - 0.758i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 + (0.338 + 0.195i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (3.44 - 5.96i)T + (-14.5 - 25.1i)T^{2} \)
31 \( 1 + (0.211 + 0.211i)T + 31iT^{2} \)
37 \( 1 + (5.84 - 1.56i)T + (32.0 - 18.5i)T^{2} \)
41 \( 1 + (-3.49 + 0.936i)T + (35.5 - 20.5i)T^{2} \)
43 \( 1 + (-2.10 + 1.21i)T + (21.5 - 37.2i)T^{2} \)
47 \( 1 + (6.84 - 6.84i)T - 47iT^{2} \)
53 \( 1 - 9.73T + 53T^{2} \)
59 \( 1 + (-5.37 - 1.44i)T + (51.0 + 29.5i)T^{2} \)
61 \( 1 + (8.38 - 4.84i)T + (30.5 - 52.8i)T^{2} \)
67 \( 1 + (-1.38 - 5.18i)T + (-58.0 + 33.5i)T^{2} \)
71 \( 1 + (-12.8 - 3.45i)T + (61.4 + 35.5i)T^{2} \)
73 \( 1 + (-5.09 + 5.09i)T - 73iT^{2} \)
79 \( 1 - 9.25T + 79T^{2} \)
83 \( 1 + (-9.33 - 9.33i)T + 83iT^{2} \)
89 \( 1 + (-1.44 - 5.37i)T + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (-3.44 + 12.8i)T + (-84.0 - 48.5i)T^{2} \)
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

−10.87632423410945601093986356865, −10.37654715505267461449344582245, −9.545326125814671466164972502957, −8.338018321672677908580215782972, −7.37595769123772671350904373026, −6.49906340975182477192502027856, −5.67866518977520742197398208475, −5.26050102006315323523291330451, −3.24567188908823558054041339545, −2.26083866175128948183630770511, 0.59948156396460870941579893628, 2.12830072911055645736472840774, 3.66620784783690516651446449736, 4.83336259991331904292013082880, 5.49969586860826382729660971432, 6.44112157780101820364910434802, 7.889268475828443785118224741615, 9.018419848461803663877482066294, 9.722936188082406058810207769250, 10.41511716939175611564226675591

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