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Properties

Label 1-44-44.7-r0-0-0
Degree $1$
Conductor $44$
Sign $0.970 + 0.242i$
Analytic cond. $0.204335$
Root an. cond. $0.204335$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)35-s + ⋯
L(s)  = 1  + (0.809 + 0.587i)3-s + (0.309 − 0.951i)5-s + (−0.809 + 0.587i)7-s + (0.309 + 0.951i)9-s + (−0.309 − 0.951i)13-s + (0.809 − 0.587i)15-s + (−0.309 + 0.951i)17-s + (−0.809 − 0.587i)19-s − 21-s − 23-s + (−0.809 − 0.587i)25-s + (−0.309 + 0.951i)27-s + (0.809 − 0.587i)29-s + (−0.309 − 0.951i)31-s + (0.309 + 0.951i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.242i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.970 + 0.242i$
Analytic conductor: \(0.204335\)
Root analytic conductor: \(0.204335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 44,\ (0:\ ),\ 0.970 + 0.242i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.9845264262 + 0.1210243728i\)
\(L(\frac12)\) \(\approx\) \(0.9845264262 + 0.1210243728i\)
\(L(1)\) \(\approx\) \(1.147835706 + 0.1032782450i\)
\(L(1)\) \(\approx\) \(1.147835706 + 0.1032782450i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 \)
good3 \( 1 + (0.809 + 0.587i)T \)
5 \( 1 + (0.309 - 0.951i)T \)
7 \( 1 + (-0.809 + 0.587i)T \)
13 \( 1 + (-0.309 - 0.951i)T \)
17 \( 1 + (-0.309 + 0.951i)T \)
19 \( 1 + (-0.809 - 0.587i)T \)
23 \( 1 - T \)
29 \( 1 + (0.809 - 0.587i)T \)
31 \( 1 + (-0.309 - 0.951i)T \)
37 \( 1 + (-0.809 + 0.587i)T \)
41 \( 1 + (0.809 + 0.587i)T \)
43 \( 1 + T \)
47 \( 1 + (0.809 + 0.587i)T \)
53 \( 1 + (0.309 + 0.951i)T \)
59 \( 1 + (0.809 - 0.587i)T \)
61 \( 1 + (-0.309 + 0.951i)T \)
67 \( 1 - T \)
71 \( 1 + (-0.309 + 0.951i)T \)
73 \( 1 + (0.809 - 0.587i)T \)
79 \( 1 + (0.309 + 0.951i)T \)
83 \( 1 + (0.309 - 0.951i)T \)
89 \( 1 + T \)
97 \( 1 + (0.309 + 0.951i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−34.4726903990881286580885193826, −33.26355481212658593813848053538, −31.96767928836159010599863382360, −30.926901046547425193632433936320, −29.73105155011965513456006951161, −29.13863537321955894967402581757, −26.98064096864290845838183301154, −26.10404482454597458155680016571, −25.27741442678676483724273209837, −23.807648060008017628316418807599, −22.64082815565675085319397970804, −21.23772218757762916387729966204, −19.772570861674500523740538334069, −18.9214696967834265214497143839, −17.752344993497623026395095692153, −16.03511470813397832786808465937, −14.412523937986258648591167946396, −13.73139700141500134458552961197, −12.24803146971115319114687098888, −10.407528118206047041616853574763, −9.148829203152485360988927369244, −7.32469845292370718990441612681, −6.46229024934163453765732537354, −3.74515999340685482127747948007, −2.28378516162309056718553418597, 2.46002988612224249827210463711, 4.24089941722468190474307235168, 5.85473018937241554376822469898, 8.133566518628181227581774950908, 9.19005223823535872004786557202, 10.31314261741727262391084873549, 12.490414261903935433789232772376, 13.446617576009133485843815830817, 15.11320098763332270269247053005, 16.032360420881753483475935149916, 17.3879257190209318568494073988, 19.25611004807683564802790105142, 20.12418446714452089172899162129, 21.350042500838695645786875926171, 22.3168751322638147246378063518, 24.1852773119330385105661647533, 25.28025677620676917364679393966, 26.081870193552683387012237569501, 27.65075393806871709494949298022, 28.44515917720875047803613506690, 29.93051930849023163243247324969, 31.417019556999409048852747305381, 32.29542511423996074260278451651, 32.87570903532075847545899958976, 34.58186347153190468479886079030

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