Properties

Label 2-44-44.35-c1-0-3
Degree $2$
Conductor $44$
Sign $0.324 + 0.945i$
Analytic cond. $0.351341$
Root an. cond. $0.592740$
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  + (0.204 − 1.39i)2-s + (−0.539 − 0.743i)3-s + (−1.91 − 0.573i)4-s + (0.529 + 1.63i)5-s + (−1.15 + 0.603i)6-s + (1.93 + 1.40i)7-s + (−1.19 + 2.56i)8-s + (0.666 − 2.05i)9-s + (2.39 − 0.407i)10-s + (−2.65 + 1.98i)11-s + (0.608 + 1.73i)12-s + (−1.52 − 0.497i)13-s + (2.36 − 2.41i)14-s + (0.925 − 1.27i)15-s + (3.34 + 2.19i)16-s + (−5.74 + 1.86i)17-s + ⋯
L(s)  = 1  + (0.144 − 0.989i)2-s + (−0.311 − 0.428i)3-s + (−0.957 − 0.286i)4-s + (0.236 + 0.729i)5-s + (−0.469 + 0.246i)6-s + (0.730 + 0.531i)7-s + (−0.422 + 0.906i)8-s + (0.222 − 0.683i)9-s + (0.755 − 0.128i)10-s + (−0.800 + 0.599i)11-s + (0.175 + 0.500i)12-s + (−0.424 − 0.137i)13-s + (0.631 − 0.646i)14-s + (0.239 − 0.328i)15-s + (0.835 + 0.549i)16-s + (−1.39 + 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.324 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.324 + 0.945i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (35, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.324 + 0.945i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.619486 - 0.442391i\)
\(L(\frac12)\) \(\approx\) \(0.619486 - 0.442391i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.204 + 1.39i)T \)
11 \( 1 + (2.65 - 1.98i)T \)
good3 \( 1 + (0.539 + 0.743i)T + (-0.927 + 2.85i)T^{2} \)
5 \( 1 + (-0.529 - 1.63i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (-1.93 - 1.40i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (1.52 + 0.497i)T + (10.5 + 7.64i)T^{2} \)
17 \( 1 + (5.74 - 1.86i)T + (13.7 - 9.99i)T^{2} \)
19 \( 1 + (-1.07 + 0.779i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 + 7.25iT - 23T^{2} \)
29 \( 1 + (0.318 - 0.439i)T + (-8.96 - 27.5i)T^{2} \)
31 \( 1 + (-7.82 - 2.54i)T + (25.0 + 18.2i)T^{2} \)
37 \( 1 + (0.299 + 0.217i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (-2.99 - 4.11i)T + (-12.6 + 38.9i)T^{2} \)
43 \( 1 + 9.45T + 43T^{2} \)
47 \( 1 + (-2.35 - 3.24i)T + (-14.5 + 44.6i)T^{2} \)
53 \( 1 + (0.765 - 2.35i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-4.14 + 5.70i)T + (-18.2 - 56.1i)T^{2} \)
61 \( 1 + (-9.41 + 3.05i)T + (49.3 - 35.8i)T^{2} \)
67 \( 1 + 4.79iT - 67T^{2} \)
71 \( 1 + (3.34 - 1.08i)T + (57.4 - 41.7i)T^{2} \)
73 \( 1 + (1.23 - 1.70i)T + (-22.5 - 69.4i)T^{2} \)
79 \( 1 + (0.797 - 2.45i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (-1.22 - 3.77i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.45T + 89T^{2} \)
97 \( 1 + (2.57 - 7.91i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.30035676842707206175264127957, −14.49290098463456369681285914499, −13.07872019983712583752725447062, −12.17439349740895176457031830641, −11.05788500286960806575559991781, −9.998171371152420931189467185418, −8.451998456557875807436910900311, −6.55718498931181233110651247168, −4.77211385045028668663464353930, −2.43970731718250358503347518610, 4.54091099639989253225350124466, 5.42921536950679246297502834054, 7.38084116022951475226201610418, 8.551602194997631995278879448563, 9.968301492202687700821637584192, 11.40939859388480613670136030010, 13.23055616689476412265281444112, 13.75946092575663644411795227874, 15.33316604893244421196485778639, 16.16692060450789056611903312425

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