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Properties

Label 2-110-55.54-c2-0-9
Degree $2$
Conductor $110$
Sign $0.116 + 0.993i$
Analytic cond. $2.99728$
Root an. cond. $1.73126$
Motivic weight $2$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 1.41·2-s − 3.43i·3-s + 2.00·4-s + (−3.90 − 3.11i)5-s − 4.86i·6-s + 2.82·7-s + 2.82·8-s − 2.81·9-s + (−5.52 − 4.41i)10-s + (−7.81 − 7.74i)11-s − 6.87i·12-s + 3.98·13-s + 4.00·14-s + (−10.7 + 13.4i)15-s + 4.00·16-s + 23.2·17-s + ⋯
L(s)  = 1  + 0.707·2-s − 1.14i·3-s + 0.500·4-s + (−0.781 − 0.623i)5-s − 0.810i·6-s + 0.404·7-s + 0.353·8-s − 0.312·9-s + (−0.552 − 0.441i)10-s + (−0.710 − 0.703i)11-s − 0.572i·12-s + 0.306·13-s + 0.285·14-s + (−0.714 + 0.895i)15-s + 0.250·16-s + 1.36·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.116 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.116 + 0.993i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (109, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ 0.116 + 0.993i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.38922 - 1.23620i\)
\(L(\frac12)\) \(\approx\) \(1.38922 - 1.23620i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 1.41T \)
5 \( 1 + (3.90 + 3.11i)T \)
11 \( 1 + (7.81 + 7.74i)T \)
good3 \( 1 + 3.43iT - 9T^{2} \)
7 \( 1 - 2.82T + 49T^{2} \)
13 \( 1 - 3.98T + 169T^{2} \)
17 \( 1 - 23.2T + 289T^{2} \)
19 \( 1 - 33.1iT - 361T^{2} \)
23 \( 1 + 2.16iT - 529T^{2} \)
29 \( 1 - 11.5iT - 841T^{2} \)
31 \( 1 - 36.7T + 961T^{2} \)
37 \( 1 - 17.4iT - 1.36e3T^{2} \)
41 \( 1 - 45.0iT - 1.68e3T^{2} \)
43 \( 1 + 63.4T + 1.84e3T^{2} \)
47 \( 1 + 46.5iT - 2.20e3T^{2} \)
53 \( 1 + 11.2iT - 2.80e3T^{2} \)
59 \( 1 + 44.7T + 3.48e3T^{2} \)
61 \( 1 - 38.8iT - 3.72e3T^{2} \)
67 \( 1 + 91.5iT - 4.48e3T^{2} \)
71 \( 1 - 54.5T + 5.04e3T^{2} \)
73 \( 1 - 56.1T + 5.32e3T^{2} \)
79 \( 1 + 101. iT - 6.24e3T^{2} \)
83 \( 1 - 15.7T + 6.88e3T^{2} \)
89 \( 1 + 28.7T + 7.92e3T^{2} \)
97 \( 1 - 76.2iT - 9.40e3T^{2} \)
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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

−13.09562779340800971944893293933, −12.23942508994659822884290422148, −11.65582985187063030501976169899, −10.24136371470189739616860653117, −8.125147201922282807779916247338, −7.87031371783802298482283448986, −6.32646947268129914989224325554, −5.08387851077189746051004884894, −3.44542111927037707933939008853, −1.33424321347456970253732503376, 2.98923183501691408625318562296, 4.26409516634650343113911881493, 5.18674699809870638591725685997, 6.94729475668135637497894314672, 8.067278446086762853168219978152, 9.723787759765641543289505161420, 10.66216565027984705589685648161, 11.44782066104145546778311860976, 12.55915497750121345068984078538, 13.87681989825659281247409265522

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