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Properties

Label 2-44-4.3-c2-0-4
Degree $2$
Conductor $44$
Sign $0.993 - 0.115i$
Analytic cond. $1.19891$
Root an. cond. $1.09494$
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  + (1.32 − 1.49i)2-s + 5.74i·3-s + (−0.463 − 3.97i)4-s + 2.62·5-s + (8.57 + 7.63i)6-s − 5.65i·7-s + (−6.55 − 4.59i)8-s − 23.9·9-s + (3.48 − 3.91i)10-s − 3.31i·11-s + (22.8 − 2.66i)12-s + 2.38·13-s + (−8.44 − 7.51i)14-s + 15.0i·15-s + (−15.5 + 3.68i)16-s + 3.93·17-s + ⋯
L(s)  = 1  + (0.664 − 0.746i)2-s + 1.91i·3-s + (−0.115 − 0.993i)4-s + 0.524·5-s + (1.42 + 1.27i)6-s − 0.807i·7-s + (−0.818 − 0.573i)8-s − 2.66·9-s + (0.348 − 0.391i)10-s − 0.301i·11-s + (1.90 − 0.221i)12-s + 0.183·13-s + (−0.603 − 0.536i)14-s + 1.00i·15-s + (−0.973 + 0.230i)16-s + 0.231·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.993 - 0.115i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.993 - 0.115i$
Analytic conductor: \(1.19891\)
Root analytic conductor: \(1.09494\)
Motivic weight: \(2\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (23, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1),\ 0.993 - 0.115i)\)

Particular Values

\(L(\frac{3}{2})\) \(\approx\) \(1.46268 + 0.0849831i\)
\(L(\frac12)\) \(\approx\) \(1.46268 + 0.0849831i\)
\(L(2)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.32 + 1.49i)T \)
11 \( 1 + 3.31iT \)
good3 \( 1 - 5.74iT - 9T^{2} \)
5 \( 1 - 2.62T + 25T^{2} \)
7 \( 1 + 5.65iT - 49T^{2} \)
13 \( 1 - 2.38T + 169T^{2} \)
17 \( 1 - 3.93T + 289T^{2} \)
19 \( 1 - 12.6iT - 361T^{2} \)
23 \( 1 + 1.98iT - 529T^{2} \)
29 \( 1 - 39.1T + 841T^{2} \)
31 \( 1 - 39.6iT - 961T^{2} \)
37 \( 1 + 32.8T + 1.36e3T^{2} \)
41 \( 1 - 20.2T + 1.68e3T^{2} \)
43 \( 1 - 11.7iT - 1.84e3T^{2} \)
47 \( 1 + 13.1iT - 2.20e3T^{2} \)
53 \( 1 - 51.4T + 2.80e3T^{2} \)
59 \( 1 - 26.8iT - 3.48e3T^{2} \)
61 \( 1 + 95.2T + 3.72e3T^{2} \)
67 \( 1 + 3.45iT - 4.48e3T^{2} \)
71 \( 1 + 18.0iT - 5.04e3T^{2} \)
73 \( 1 + 54.0T + 5.32e3T^{2} \)
79 \( 1 + 97.1iT - 6.24e3T^{2} \)
83 \( 1 + 118. iT - 6.88e3T^{2} \)
89 \( 1 - 21.4T + 7.92e3T^{2} \)
97 \( 1 - 0.0566T + 9.40e3T^{2} \)
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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

−15.58198128533679533495904947357, −14.37934848512353612788349320193, −13.75306899593276024367392214557, −11.88319086688610696560000515116, −10.54206913532375081302542429679, −10.17905139549572592481996070693, −8.899414113513106745613947621006, −5.88139180229748829257229538880, −4.58633495788027556362846224907, −3.36013491610043357727506793212, 2.46458505002311888245559443197, 5.59269273214546809936320537974, 6.54635923680101136461079910947, 7.74216055442115372655554386656, 8.892929938251445041988460410610, 11.65762924830750074629237092393, 12.45718270146370857175029251236, 13.40399356105791350939571787222, 14.14490254061819232169985507647, 15.41935882604722235966683457896

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