↑

Properties

Label 2-55-11.3-c1-0-3
Degree $2$
Conductor $55$
Sign $0.564 + 0.825i$
Analytic cond. $0.439177$
Root an. cond. $0.662704$
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  + (0.596 − 0.433i)2-s + (−0.868 − 2.67i)3-s + (−0.449 + 1.38i)4-s + (0.809 + 0.587i)5-s + (−1.67 − 1.21i)6-s + (0.318 − 0.980i)7-s + (0.787 + 2.42i)8-s + (−3.96 + 2.88i)9-s + 0.737·10-s + (1.93 + 2.69i)11-s + 4.09·12-s + (−2.79 + 2.02i)13-s + (−0.235 − 0.723i)14-s + (0.868 − 2.67i)15-s + (−0.834 − 0.606i)16-s + (−1.94 − 1.40i)17-s + ⋯
L(s)  = 1  + (0.421 − 0.306i)2-s + (−0.501 − 1.54i)3-s + (−0.224 + 0.692i)4-s + (0.361 + 0.262i)5-s + (−0.684 − 0.497i)6-s + (0.120 − 0.370i)7-s + (0.278 + 0.857i)8-s + (−1.32 + 0.961i)9-s + 0.233·10-s + (0.583 + 0.811i)11-s + 1.18·12-s + (−0.773 + 0.562i)13-s + (−0.0628 − 0.193i)14-s + (0.224 − 0.690i)15-s + (−0.208 − 0.151i)16-s + (−0.470 − 0.341i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.564 + 0.825i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(55\)    =    \(5 \cdot 11\)
Sign: $0.564 + 0.825i$
Analytic conductor: \(0.439177\)
Root analytic conductor: \(0.662704\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{55} (36, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 55,\ (\ :1/2),\ 0.564 + 0.825i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.796395 - 0.420186i\)
\(L(\frac12)\) \(\approx\) \(0.796395 - 0.420186i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad5 \( 1 + (-0.809 - 0.587i)T \)
11 \( 1 + (-1.93 - 2.69i)T \)
good2 \( 1 + (-0.596 + 0.433i)T + (0.618 - 1.90i)T^{2} \)
3 \( 1 + (0.868 + 2.67i)T + (-2.42 + 1.76i)T^{2} \)
7 \( 1 + (-0.318 + 0.980i)T + (-5.66 - 4.11i)T^{2} \)
13 \( 1 + (2.79 - 2.02i)T + (4.01 - 12.3i)T^{2} \)
17 \( 1 + (1.94 + 1.40i)T + (5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.36 + 7.29i)T + (-15.3 + 11.1i)T^{2} \)
23 \( 1 - 2.45T + 23T^{2} \)
29 \( 1 + (1.83 - 5.66i)T + (-23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.98 + 2.16i)T + (9.57 - 29.4i)T^{2} \)
37 \( 1 + (-1.84 + 5.66i)T + (-29.9 - 21.7i)T^{2} \)
41 \( 1 + (-1.21 - 3.74i)T + (-33.1 + 24.0i)T^{2} \)
43 \( 1 + 7.64T + 43T^{2} \)
47 \( 1 + (-1.80 - 5.55i)T + (-38.0 + 27.6i)T^{2} \)
53 \( 1 + (-9.58 + 6.96i)T + (16.3 - 50.4i)T^{2} \)
59 \( 1 + (-0.910 + 2.80i)T + (-47.7 - 34.6i)T^{2} \)
61 \( 1 + (2.00 + 1.45i)T + (18.8 + 58.0i)T^{2} \)
67 \( 1 + 6.14T + 67T^{2} \)
71 \( 1 + (1.63 + 1.18i)T + (21.9 + 67.5i)T^{2} \)
73 \( 1 + (0.255 - 0.785i)T + (-59.0 - 42.9i)T^{2} \)
79 \( 1 + (-9.77 + 7.09i)T + (24.4 - 75.1i)T^{2} \)
83 \( 1 + (1.30 + 0.946i)T + (25.6 + 78.9i)T^{2} \)
89 \( 1 - 8.16T + 89T^{2} \)
97 \( 1 + (1.97 - 1.43i)T + (29.9 - 92.2i)T^{2} \)
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

−14.70276550900625826788232814992, −13.59370688928729702329316660612, −12.93088582731995426365198531894, −11.97909322681601117888362037108, −11.11746631921388414969948292222, −9.071398378020895719399577538642, −7.40963601379384653372287823625, −6.75194129837842569168796799919, −4.75197818114324137406326172222, −2.31346873743097894295491064814, 4.03154735665659843412821333460, 5.28116635919281508337924941740, 6.12051626079621319602845311024, 8.727229894635258941157291925049, 9.888524781409451832097569030107, 10.56931121538238887032405184149, 11.97701002392100713111873507865, 13.56925990542647407919013103904, 14.81654747253510764756256985637, 15.26316081571501091390682351960

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