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Properties

Label 2-110-5.4-c3-0-3
Degree $2$
Conductor $110$
Sign $-0.987 + 0.159i$
Analytic cond. $6.49021$
Root an. cond. $2.54758$
Motivic weight $3$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2i·2-s + 5.97i·3-s − 4·4-s + (1.78 + 11.0i)5-s − 11.9·6-s + 8.09i·7-s − 8i·8-s − 8.70·9-s + (−22.0 + 3.56i)10-s − 11·11-s − 23.9i·12-s − 14.2i·13-s − 16.1·14-s + (−65.9 + 10.6i)15-s + 16·16-s − 19.5i·17-s + ⋯
L(s)  = 1  + 0.707i·2-s + 1.15i·3-s − 0.5·4-s + (0.159 + 0.987i)5-s − 0.813·6-s + 0.437i·7-s − 0.353i·8-s − 0.322·9-s + (−0.698 + 0.112i)10-s − 0.301·11-s − 0.575i·12-s − 0.303i·13-s − 0.309·14-s + (−1.13 + 0.183i)15-s + 0.250·16-s − 0.279i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.987 + 0.159i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $-0.987 + 0.159i$
Analytic conductor: \(6.49021\)
Root analytic conductor: \(2.54758\)
Motivic weight: \(3\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (89, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :3/2),\ -0.987 + 0.159i)\)

Particular Values

\(L(2)\) \(\approx\) \(0.107148 - 1.33674i\)
\(L(\frac12)\) \(\approx\) \(0.107148 - 1.33674i\)
\(L(\frac{5}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - 2iT \)
5 \( 1 + (-1.78 - 11.0i)T \)
11 \( 1 + 11T \)
good3 \( 1 - 5.97iT - 27T^{2} \)
7 \( 1 - 8.09iT - 343T^{2} \)
13 \( 1 + 14.2iT - 2.19e3T^{2} \)
17 \( 1 + 19.5iT - 4.91e3T^{2} \)
19 \( 1 + 3.12T + 6.85e3T^{2} \)
23 \( 1 + 57.9iT - 1.21e4T^{2} \)
29 \( 1 - 110.T + 2.43e4T^{2} \)
31 \( 1 + 53.2T + 2.97e4T^{2} \)
37 \( 1 - 380. iT - 5.06e4T^{2} \)
41 \( 1 - 217.T + 6.89e4T^{2} \)
43 \( 1 - 200. iT - 7.95e4T^{2} \)
47 \( 1 - 434. iT - 1.03e5T^{2} \)
53 \( 1 + 550. iT - 1.48e5T^{2} \)
59 \( 1 + 199.T + 2.05e5T^{2} \)
61 \( 1 - 690.T + 2.26e5T^{2} \)
67 \( 1 - 622. iT - 3.00e5T^{2} \)
71 \( 1 - 209.T + 3.57e5T^{2} \)
73 \( 1 + 40.3iT - 3.89e5T^{2} \)
79 \( 1 + 765.T + 4.93e5T^{2} \)
83 \( 1 + 525. iT - 5.71e5T^{2} \)
89 \( 1 + 788.T + 7.04e5T^{2} \)
97 \( 1 + 1.21e3iT - 9.12e5T^{2} \)
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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

−14.15142117769613108708649644788, −12.88206852584101890831704541246, −11.36959551679025542256666568578, −10.31217210511205256516574935489, −9.610439441442113490985729133782, −8.329146726583324320297094949268, −6.97491139083390150124931139151, −5.74212747285037852791535785749, −4.49204720840690416528601927506, −3.00804596705165977852824501247, 0.76522361828973976535555559743, 2.04784672122505009953211406151, 4.13781190820268967507683573226, 5.64067471452063037376209947010, 7.18922274923349606515282628944, 8.270067553567689496446860425699, 9.387499761407226317227644549013, 10.63322757192260229022451771668, 11.93973332639251238627195789299, 12.62454339088612273495962912709

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