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Properties

Label 2-110-11.10-c2-0-1
Degree $2$
Conductor $110$
Sign $0.752 - 0.658i$
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  − 1.41i·2-s − 5.11·3-s − 2.00·4-s + 2.23·5-s + 7.23i·6-s + 6.78i·7-s + 2.82i·8-s + 17.1·9-s − 3.16i·10-s + (7.23 + 8.28i)11-s + 10.2·12-s − 9.87i·13-s + 9.59·14-s − 11.4·15-s + 4.00·16-s + 29.1i·17-s + ⋯
L(s)  = 1  − 0.707i·2-s − 1.70·3-s − 0.500·4-s + 0.447·5-s + 1.20i·6-s + 0.968i·7-s + 0.353i·8-s + 1.90·9-s − 0.316i·10-s + (0.658 + 0.752i)11-s + 0.852·12-s − 0.759i·13-s + 0.685·14-s − 0.762·15-s + 0.250·16-s + 1.71i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 110 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.752 - 0.658i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(110\)    =    \(2 \cdot 5 \cdot 11\)
Sign: $0.752 - 0.658i$
Analytic conductor: \(2.99728\)
Root analytic conductor: \(1.73126\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{110} (21, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 110,\ (\ :1),\ 0.752 - 0.658i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(0.632158 + 0.237326i\)
\(L(\frac12)\) \(\approx\) \(0.632158 + 0.237326i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + 1.41iT \)
5 \( 1 - 2.23T \)
11 \( 1 + (-7.23 - 8.28i)T \)
good3 \( 1 + 5.11T + 9T^{2} \)
7 \( 1 - 6.78iT - 49T^{2} \)
13 \( 1 + 9.87iT - 169T^{2} \)
17 \( 1 - 29.1iT - 289T^{2} \)
19 \( 1 - 27.1iT - 361T^{2} \)
23 \( 1 + 35.0T + 529T^{2} \)
29 \( 1 + 21.3iT - 841T^{2} \)
31 \( 1 - 21.1T + 961T^{2} \)
37 \( 1 + 7.43T + 1.36e3T^{2} \)
41 \( 1 - 44.2iT - 1.68e3T^{2} \)
43 \( 1 - 15.2iT - 1.84e3T^{2} \)
47 \( 1 + 0.490T + 2.20e3T^{2} \)
53 \( 1 - 40.6T + 2.80e3T^{2} \)
59 \( 1 + 15.7T + 3.48e3T^{2} \)
61 \( 1 + 36.9iT - 3.72e3T^{2} \)
67 \( 1 - 98.6T + 4.48e3T^{2} \)
71 \( 1 + 114.T + 5.04e3T^{2} \)
73 \( 1 - 106. iT - 5.32e3T^{2} \)
79 \( 1 + 35.4iT - 6.24e3T^{2} \)
83 \( 1 + 119. iT - 6.88e3T^{2} \)
89 \( 1 + 121.T + 7.92e3T^{2} \)
97 \( 1 - 124.T + 9.40e3T^{2} \)
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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

−12.91708867217356877618246390110, −12.27104598653451339619855658167, −11.67242208834125379947629304943, −10.36935861637125738230330244139, −9.883054384467251706511815341229, −8.190650332041218273083715553744, −6.21303994193361696405042043882, −5.67529963779538838319476507960, −4.19811796523380033403782218649, −1.70528011818008952079499287072, 0.64569573679125452796676559794, 4.30859078746593483194060909847, 5.39066651131387574500866338023, 6.56456283808843862400351075986, 7.17234519280633547051228521187, 9.111475095326106956842181289682, 10.23985804954212388583145050488, 11.28288044038110673208778059987, 12.06623101342770511517153778627, 13.56329032933638785586038425563

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