↑

Properties

Label 2-110-11.4-c1-0-1
Degree $2$
Conductor $110$
Sign $0.751 - 0.659i$
Analytic cond. $0.878354$
Root an. cond. $0.937205$
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  + (0.809 + 0.587i)2-s + (−0.118 + 0.363i)3-s + (0.309 + 0.951i)4-s + (0.809 − 0.587i)5-s + (−0.309 + 0.224i)6-s + (−0.309 + 0.951i)8-s + (2.30 + 1.67i)9-s + 10-s + (−3.30 + 0.224i)11-s − 0.381·12-s + (−5.23 − 3.80i)13-s + (0.118 + 0.363i)15-s + (−0.809 + 0.587i)16-s + (4.11 − 2.99i)17-s + (0.881 + 2.71i)18-s + (1.19 − 3.66i)19-s + ⋯
L(s)  = 1  + (0.572 + 0.415i)2-s + (−0.0681 + 0.209i)3-s + (0.154 + 0.475i)4-s + (0.361 − 0.262i)5-s + (−0.126 + 0.0916i)6-s + (−0.109 + 0.336i)8-s + (0.769 + 0.559i)9-s + 0.316·10-s + (−0.997 + 0.0676i)11-s − 0.110·12-s + (−1.45 − 1.05i)13-s + (0.0304 + 0.0937i)15-s + (−0.202 + 0.146i)16-s + (0.998 − 0.725i)17-s + (0.207 + 0.639i)18-s + (0.273 − 0.840i)19-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (81, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.27359 + 0.479265i\)
\(L(\frac12)\) \(\approx\) \(1.27359 + 0.479265i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.809 - 0.587i)T \)
5 \( 1 + (-0.809 + 0.587i)T \)
11 \( 1 + (3.30 - 0.224i)T \)
good3 \( 1 + (0.118 - 0.363i)T + (-2.42 - 1.76i)T^{2} \)
7 \( 1 + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (5.23 + 3.80i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (-4.11 + 2.99i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.19 + 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 6T + 23T^{2} \)
29 \( 1 + (-1.47 - 4.53i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (-3 - 2.17i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.618 + 1.90i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (1.5 - 4.61i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 2.85T + 43T^{2} \)
47 \( 1 + (3.47 - 10.6i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-2.85 - 2.07i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-3.57 - 10.9i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (-6.23 + 4.53i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 + 0.618T + 67T^{2} \)
71 \( 1 + (-3.85 + 2.80i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (-0.354 - 1.08i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (5.85 + 4.25i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-10.0 + 7.27i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 4.09T + 89T^{2} \)
97 \( 1 + (-6.16 - 4.47i)T + (29.9 + 92.2i)T^{2} \)
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

−13.73391452145476024895067148773, −12.86713089025695066018477637692, −12.05506198831344428589950581182, −10.46222840186804594494285796650, −9.740094405059532301928704756411, −8.006191806351943963938259972958, −7.21336146360984947067669365029, −5.44709701688329747017714584992, −4.77789797796261333236016272799, −2.75699032752421946147303638615, 2.13842502971745984363604282403, 3.94314863791427471019210633483, 5.42485017875667382003178062924, 6.68019280906200264330130118684, 7.896920547700136663311500835972, 9.851033880046425376611521368883, 10.15178375608840098602327926718, 11.84872826613371982670096117731, 12.38585377805247943848938960898, 13.51763371679693611936319606322

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