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Properties

Label 2-110-55.28-c1-0-2
Degree $2$
Conductor $110$
Sign $0.999 - 0.0126i$
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 + (0.256 + 1.61i)3-s + (−0.587 − 0.809i)4-s + (1.69 + 1.45i)5-s + (1.55 + 0.506i)6-s + (−0.753 − 0.119i)7-s + (−0.987 + 0.156i)8-s + (0.299 − 0.0972i)9-s + (2.06 − 0.849i)10-s + (−1.49 − 2.95i)11-s + (1.15 − 1.15i)12-s + (−0.760 − 0.387i)13-s + (−0.448 + 0.616i)14-s + (−1.92 + 3.11i)15-s + (−0.309 + 0.951i)16-s + (−1.29 + 0.657i)17-s + ⋯
L(s)  = 1  + (0.321 − 0.630i)2-s + (0.148 + 0.934i)3-s + (−0.293 − 0.404i)4-s + (0.758 + 0.651i)5-s + (0.636 + 0.206i)6-s + (−0.284 − 0.0450i)7-s + (−0.349 + 0.0553i)8-s + (0.0997 − 0.0324i)9-s + (0.654 − 0.268i)10-s + (−0.451 − 0.892i)11-s + (0.334 − 0.334i)12-s + (−0.210 − 0.107i)13-s + (−0.119 + 0.164i)14-s + (−0.496 + 0.805i)15-s + (−0.0772 + 0.237i)16-s + (−0.312 + 0.159i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0126i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0126i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.999 - 0.0126i$
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: \((2,\ 110,\ (\ :1/2),\ 0.999 - 0.0126i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.28045 + 0.00807911i\)
\(L(\frac12)\) \(\approx\) \(1.28045 + 0.00807911i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.69 - 1.45i)T \)
11 \( 1 + (1.49 + 2.95i)T \)
good3 \( 1 + (-0.256 - 1.61i)T + (-2.85 + 0.927i)T^{2} \)
7 \( 1 + (0.753 + 0.119i)T + (6.65 + 2.16i)T^{2} \)
13 \( 1 + (0.760 + 0.387i)T + (7.64 + 10.5i)T^{2} \)
17 \( 1 + (1.29 - 0.657i)T + (9.99 - 13.7i)T^{2} \)
19 \( 1 + (4.09 + 2.97i)T + (5.87 + 18.0i)T^{2} \)
23 \( 1 + (1.65 + 1.65i)T + 23iT^{2} \)
29 \( 1 + (0.552 - 0.401i)T + (8.96 - 27.5i)T^{2} \)
31 \( 1 + (1.08 + 3.34i)T + (-25.0 + 18.2i)T^{2} \)
37 \( 1 + (1.26 - 8.01i)T + (-35.1 - 11.4i)T^{2} \)
41 \( 1 + (-7.30 + 10.0i)T + (-12.6 - 38.9i)T^{2} \)
43 \( 1 + (7.61 - 7.61i)T - 43iT^{2} \)
47 \( 1 + (-12.5 + 1.98i)T + (44.6 - 14.5i)T^{2} \)
53 \( 1 + (-3.72 + 7.31i)T + (-31.1 - 42.8i)T^{2} \)
59 \( 1 + (-0.254 - 0.350i)T + (-18.2 + 56.1i)T^{2} \)
61 \( 1 + (-12.0 - 3.91i)T + (49.3 + 35.8i)T^{2} \)
67 \( 1 + (3.82 - 3.82i)T - 67iT^{2} \)
71 \( 1 + (2.45 - 7.55i)T + (-57.4 - 41.7i)T^{2} \)
73 \( 1 + (0.179 - 1.13i)T + (-69.4 - 22.5i)T^{2} \)
79 \( 1 + (-3.11 - 9.59i)T + (-63.9 + 46.4i)T^{2} \)
83 \( 1 + (0.108 + 0.212i)T + (-48.7 + 67.1i)T^{2} \)
89 \( 1 - 11.2iT - 89T^{2} \)
97 \( 1 + (12.6 + 6.45i)T + (57.0 + 78.4i)T^{2} \)
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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

−13.59517584398137496180490933772, −12.83174449728444091674432520937, −11.25501212753385586927680185096, −10.47706678833000787961313027160, −9.771494551860266179334534607667, −8.669233118919703737310699810902, −6.70461595636324576281937467807, −5.41809437894301104002144961976, −3.97186453181384143630987850519, −2.63027926661264606270276793065, 2.08852039325828745864020529314, 4.48865527530951970505472553136, 5.85068525115254047686373548393, 6.95354935087037357364341338520, 7.966776030448576643360422712566, 9.185242642040558651579534174604, 10.31302828739094360696929116096, 12.25673806352418367185086026774, 12.77380566848914963920251571759, 13.53185756604076712860674059886

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