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Properties

Label 2-110-55.28-c1-0-3
Degree $2$
Conductor $110$
Sign $0.204 + 0.978i$
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  + (0.453 − 0.891i)2-s + (−0.0342 − 0.216i)3-s + (−0.587 − 0.809i)4-s + (−0.388 − 2.20i)5-s + (−0.208 − 0.0677i)6-s + (0.104 + 0.0165i)7-s + (−0.987 + 0.156i)8-s + (2.80 − 0.912i)9-s + (−2.13 − 0.653i)10-s + (−0.733 + 3.23i)11-s + (−0.154 + 0.154i)12-s + (1.77 + 0.904i)13-s + (0.0623 − 0.0858i)14-s + (−0.463 + 0.159i)15-s + (−0.309 + 0.951i)16-s + (−4.77 + 2.43i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (−0.0197 − 0.124i)3-s + (−0.293 − 0.404i)4-s + (−0.173 − 0.984i)5-s + (−0.0850 − 0.0276i)6-s + (0.0396 + 0.00627i)7-s + (−0.349 + 0.0553i)8-s + (0.935 − 0.304i)9-s + (−0.676 − 0.206i)10-s + (−0.221 + 0.975i)11-s + (−0.0447 + 0.0447i)12-s + (0.492 + 0.250i)13-s + (0.0166 − 0.0229i)14-s + (−0.119 + 0.0412i)15-s + (−0.0772 + 0.237i)16-s + (−1.15 + 0.590i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.204 + 0.978i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.204 + 0.978i$
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: \((2,\ 110,\ (\ :1/2),\ 0.204 + 0.978i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.902918 - 0.734059i\)
\(L(\frac12)\) \(\approx\) \(0.902918 - 0.734059i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.388 + 2.20i)T \)
11 \( 1 + (0.733 - 3.23i)T \)
good3 \( 1 + (0.0342 + 0.216i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (-0.104 - 0.0165i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (-1.77 - 0.904i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (4.77 - 2.43i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (-3.72 - 2.70i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.26 + 1.26i)T + 23iT^{2} \)
29 \( 1 + (-3.14 + 2.28i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (-2.80 - 8.62i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (-1.42 + 9.02i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-0.361 + 0.498i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (2.10 - 2.10i)T - 43iT^{2} \)
47 \( 1 + (8.76 - 1.38i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.65 + 7.17i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (6.35 + 8.75i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (0.692 + 0.225i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (-1.43 + 1.43i)T - 67iT^{2} \)
71 \( 1 + (-0.219 + 0.674i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (2.08 - 13.1i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-2.31 - 7.12i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (2.58 + 5.07i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 0.464iT - 89T^{2} \)
97 \( 1 + (1.58 + 0.809i)T + (57.0 + 78.4i)T^{2} \)
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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

−13.04443755933509969749634313001, −12.57432365453128199201468309066, −11.60118604752816502530093427377, −10.25387552369919496834416805450, −9.340771309958661783741297316994, −8.125080730468029132393091400651, −6.61322688558989367182453944076, −4.96183970463473833742727916623, −3.97859914882634628801693708093, −1.65951359393501332079198609665, 3.07808894231425187121286157192, 4.57855079704077545607995015222, 6.12637061277398760066640372912, 7.14456188016796856438080796575, 8.194976676782776952731250880842, 9.640743704663110522740229640488, 10.84874643068370234985474703157, 11.70826500154213585324288611814, 13.42896277457308171735783855957, 13.66195782146405057450205630701

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