Properties

Label 1-44-44.15-r1-0-0
Degree $1$
Conductor $44$
Sign $0.999 + 0.0237i$
Analytic cond. $4.72845$
Root an. cond. $4.72845$
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

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.999 + 0.0237i$
Analytic conductor: \(4.72845\)
Root analytic conductor: \(4.72845\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (15, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 44,\ (1:\ ),\ 0.999 + 0.0237i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(2.120847620 + 0.02521161975i\)
\(L(\frac12)\) \(\approx\) \(2.120847620 + 0.02521161975i\)
\(L(1)\) \(\approx\) \(1.532210475 + 0.03643351569i\)
\(L(1)\) \(\approx\) \(1.532210475 + 0.03643351569i\)

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

−34.39878985917838826041223035158, −32.99043700794769474473874003342, −31.69028417376710656480657364379, −30.40810113878920278753504308418, −30.11690357671091393808131912951, −28.39240728581687361433391055341, −26.918052485463230726723885058443, −25.84327288265596404624485793559, −24.89744075359983059199875876713, −23.7299652207707264093112472389, −22.18061683998117381433338267447, −20.99187035358729752890035162497, −19.67005016983163275871124655102, −18.3729178796391340823330463949, −17.73702413900077357848978999070, −15.36964054976571955614179883777, −14.533777685953885593950040618340, −13.35250636932079065766103042274, −11.81437440968172593052911125476, −10.23305919263583406651242412549, −8.573244594583036803292576545267, −7.39337960307907603264948281719, −5.80291389716235351742311101179, −3.35313140344065141300371893875, −1.89128691692274294189259062066, 1.70517376550139498620544639056, 3.94540582846430890848593247677, 5.186039188173345398219983388094, 7.61765554963716825429929031663, 8.8657363571163944846817778988, 10.01247109345651253294879116836, 11.7005337041833146223390914089, 13.54530925416870274614199620723, 14.30681670669714702161839578699, 15.97947976546516004486151035897, 16.913362738862897498169780715053, 18.584753854340232542888776550345, 20.34039671226209999924523330444, 20.6693684829597290560639306736, 21.97694340279532990810005258908, 23.821340555279006647927917085414, 24.77819160688995790915976734729, 26.01505327701698906833488030972, 27.17145833024310271749714268253, 28.12874271153399275142878040697, 29.59270681672405290588600650773, 31.01837581008879917284198753925, 31.84083252375502723146456049261, 33.09471834866611544305101885476, 33.72335995347242087506035851772

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