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Properties

Label 2-44-44.35-c1-0-0
Degree $2$
Conductor $44$
Sign $0.337 - 0.941i$
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  + (−0.656 + 1.25i)2-s + (0.539 + 0.743i)3-s + (−1.13 − 1.64i)4-s + (0.529 + 1.63i)5-s + (−1.28 + 0.188i)6-s + (−1.93 − 1.40i)7-s + (2.80 − 0.344i)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 + (3.02 − 1.49i)14-s + (−0.925 + 1.27i)15-s + (−1.41 + 3.74i)16-s + (−5.74 + 1.86i)17-s + ⋯
L(s)  = 1  + (−0.464 + 0.885i)2-s + (0.311 + 0.428i)3-s + (−0.568 − 0.822i)4-s + (0.236 + 0.729i)5-s + (−0.524 + 0.0768i)6-s + (−0.730 − 0.531i)7-s + (0.992 − 0.121i)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 + (0.809 − 0.400i)14-s + (−0.239 + 0.328i)15-s + (−0.352 + 0.935i)16-s + (−1.39 + 0.453i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.337 - 0.941i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.337 - 0.941i$
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: \((2,\ 44,\ (\ :1/2),\ 0.337 - 0.941i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.554108 + 0.389961i\)
\(L(\frac12)\) \(\approx\) \(0.554108 + 0.389961i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.656 - 1.25i)T \)
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

−16.09140799002208181985822217134, −15.08426162762916254128155652213, −14.26936801002421279349734497950, −13.09388504003339997455749294748, −10.99550523131518665160506964041, −9.816421525742022861883093186821, −8.946846730296569581642798795020, −7.13388381996385442067853946073, −6.17213050864530239159204586855, −3.88494080891157511628563271071, 2.26086986162037297193259723631, 4.60794759224348415695938603222, 7.02881281108177739538437781557, 8.709563467419935866322606387008, 9.431743560750599217369205200876, 10.93928357898010026922215466966, 12.52240619518201760232488876048, 12.88011732796410131784448733860, 14.22692611036656743720020848638, 16.04659872861683193819876034637

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