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Properties

Label 2-110-55.9-c1-0-2
Degree $2$
Conductor $110$
Sign $0.775 - 0.630i$
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

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.951 + 0.309i)2-s + (0.363 + 0.5i)3-s + (0.809 + 0.587i)4-s + (−1.38 + 1.75i)5-s + (0.190 + 0.587i)6-s + (0.175 − 0.241i)7-s + (0.587 + 0.809i)8-s + (0.809 − 2.48i)9-s + (−1.85 + 1.24i)10-s + (1.81 − 2.77i)11-s + 0.618i·12-s + (−5.43 − 1.76i)13-s + (0.241 − 0.175i)14-s + (−1.38 − 0.0547i)15-s + (0.309 + 0.951i)16-s + (5.36 − 1.74i)17-s + ⋯
L(s)  = 1  + (0.672 + 0.218i)2-s + (0.209 + 0.288i)3-s + (0.404 + 0.293i)4-s + (−0.619 + 0.785i)5-s + (0.0779 + 0.239i)6-s + (0.0663 − 0.0913i)7-s + (0.207 + 0.286i)8-s + (0.269 − 0.829i)9-s + (−0.588 + 0.392i)10-s + (0.548 − 0.836i)11-s + 0.178i·12-s + (−1.50 − 0.490i)13-s + (0.0645 − 0.0469i)14-s + (−0.356 − 0.0141i)15-s + (0.0772 + 0.237i)16-s + (1.30 − 0.422i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.775 - 0.630i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.775 - 0.630i$
Analytic conductor: \(0.878354\)
Root analytic conductor: \(0.937205\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1/2),\ 0.775 - 0.630i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.34485 + 0.477705i\)
\(L(\frac12)\) \(\approx\) \(1.34485 + 0.477705i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.951 - 0.309i)T \)
5 \( 1 + (1.38 - 1.75i)T \)
11 \( 1 + (-1.81 + 2.77i)T \)
good3 \( 1 + (-0.363 - 0.5i)T + (-0.927 + 2.85i)T^{2} \)
7 \( 1 + (-0.175 + 0.241i)T + (-2.16 - 6.65i)T^{2} \)
13 \( 1 + (5.43 + 1.76i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (-5.36 + 1.74i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (6.17 - 4.48i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 2.78iT - 23T^{2} \)
29 \( 1 + (2.62 + 1.90i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.56 - 4.80i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (-1.91 + 2.63i)T + (-11.4 - 35.1i)T^{2} \)
41 \( 1 + (0.281 - 0.204i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 - 1.38iT - 43T^{2} \)
47 \( 1 + (-2.96 - 4.08i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (-2.34 - 0.761i)T + (42.8 + 31.1i)T^{2} \)
59 \( 1 + (2.94 + 2.13i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (2.56 + 7.90i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 + (-3.48 - 10.7i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-1.71 + 2.35i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (-0.799 + 2.45i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-2.68 + 0.871i)T + (67.1 - 48.7i)T^{2} \)
89 \( 1 - 5.09T + 89T^{2} \)
97 \( 1 + (14.9 + 4.86i)T + (78.4 + 57.0i)T^{2} \)
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   \(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−14.28721548703173609447348571829, −12.56724369020596071926478140731, −11.95332308425790039339416159840, −10.73176588022403302573221929248, −9.650747521865961721815946680494, −8.069301772137396967227761523816, −7.06296096852740025998473958472, −5.82416174578938191238133550944, −4.11626979663920680151937477117, −3.09307460082773235382088691350, 2.12608863689968860273047805965, 4.23198207857379323715361728917, 5.09585589991029180098013146158, 6.94880252705644666956739085783, 7.905365106927861104841398065689, 9.292028967122852569688174116591, 10.53726950319324581029274327541, 11.91155733117422050196663965131, 12.49122189601102087958686879284, 13.34069272806421825353523113482

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