↑

Properties

Label 2-110-55.28-c1-0-0
Degree $2$
Conductor $110$
Sign $-0.629 - 0.776i$
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  + (−0.453 + 0.891i)2-s + (0.322 + 2.03i)3-s + (−0.587 − 0.809i)4-s + (−1.79 + 1.33i)5-s + (−1.95 − 0.636i)6-s + (0.386 + 0.0611i)7-s + (0.987 − 0.156i)8-s + (−1.18 + 0.384i)9-s + (−0.373 − 2.20i)10-s + (−3.24 − 0.696i)11-s + (1.45 − 1.45i)12-s + (5.11 + 2.60i)13-s + (−0.229 + 0.316i)14-s + (−3.29 − 3.22i)15-s + (−0.309 + 0.951i)16-s + (0.440 − 0.224i)17-s + ⋯
L(s)  = 1  + (−0.321 + 0.630i)2-s + (0.186 + 1.17i)3-s + (−0.293 − 0.404i)4-s + (−0.802 + 0.596i)5-s + (−0.799 − 0.259i)6-s + (0.145 + 0.0231i)7-s + (0.349 − 0.0553i)8-s + (−0.394 + 0.128i)9-s + (−0.118 − 0.697i)10-s + (−0.977 − 0.210i)11-s + (0.420 − 0.420i)12-s + (1.41 + 0.723i)13-s + (−0.0614 + 0.0845i)14-s + (−0.850 − 0.831i)15-s + (−0.0772 + 0.237i)16-s + (0.106 − 0.0544i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.629 - 0.776i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.629 - 0.776i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (83, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ -0.629 - 0.776i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.355182 + 0.745268i\)
\(L(\frac12)\) \(\approx\) \(0.355182 + 0.745268i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.453 - 0.891i)T \)
5 \( 1 + (1.79 - 1.33i)T \)
11 \( 1 + (3.24 + 0.696i)T \)
good3 \( 1 + (-0.322 - 2.03i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.386 - 0.0611i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (-5.11 - 2.60i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (-0.440 + 0.224i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-2.61 - 1.90i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (-1.05 - 1.05i)T + 23iT^{2} \)
29 \( 1 + (-6.85 + 4.97i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.58 + 4.87i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.45 + 9.20i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (3.50 - 4.82i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (3.86 - 3.86i)T - 43iT^{2} \)
47 \( 1 + (3.76 - 0.596i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (0.830 - 1.62i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (1.77 + 2.44i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-7.86 - 2.55i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (10.6 - 10.6i)T - 67iT^{2} \)
71 \( 1 + (-3.83 + 11.8i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (-1.89 + 11.9i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (2.96 + 9.12i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (-2.69 - 5.29i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 + 9.83iT - 89T^{2} \)
97 \( 1 + (-4.79 - 2.44i)T + (57.0 + 78.4i)T^{2} \)
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

−14.42390825088742606189983837773, −13.32435613008494854541596273902, −11.56051707023758991091214338321, −10.72248600918111622190624489515, −9.792288699518951202882198719502, −8.593201830696596124442542255446, −7.64459072943214194545313325603, −6.17922187120906596518292648799, −4.64537779815049752486868191500, −3.46539519465211544068246343191, 1.19153030309280689771235093034, 3.18252937433021844495566737287, 5.01451373623015347336511191892, 6.91721791001739654631168202060, 8.087206889794234773354258785032, 8.558437719170440091344606144510, 10.30053712230434034939988621356, 11.36926284439333369033394158931, 12.41033232757416787063067274559, 13.02599423431834045503876218766

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