Properties

Label 1-44-44.43-r0-0-0
Degree $1$
Conductor $44$
Sign $1$
Analytic cond. $0.204335$
Root an. cond. $0.204335$
Motivic weight $0$
Arithmetic yes
Rational yes
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯
L(s)  = 1  − 3-s + 5-s + 7-s + 9-s − 13-s − 15-s − 17-s + 19-s − 21-s − 23-s + 25-s − 27-s − 29-s − 31-s + 35-s + 37-s + 39-s − 41-s + 43-s + 45-s − 47-s + 49-s + 51-s + 53-s − 57-s − 59-s − 61-s + ⋯

Functional equation

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

Invariants

Degree: \(1\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $1$
Analytic conductor: \(0.204335\)
Root analytic conductor: \(0.204335\)
Motivic weight: \(0\)
Rational: yes
Arithmetic: yes
Character: $\chi_{44} (43, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((1,\ 44,\ (0:\ ),\ 1)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7600622838\)
\(L(\frac12)\) \(\approx\) \(0.7600622838\)
\(L(1)\) \(\approx\) \(0.9024906449\)
\(L(1)\) \(\approx\) \(0.9024906449\)

Euler product

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

−34.15346729684272106043380578223, −33.60625213311540800317874948376, −32.488723456435195652185644207532, −30.7837343166950765025407660372, −29.60681027223561542267337147912, −28.77893942445047647072544291977, −27.60237248544315232420294111024, −26.442423451163784130338143627684, −24.681483218497409008454757604732, −24.04471027020575246774989117385, −22.32883476081719799605147614760, −21.66563972555785204780870557439, −20.30638125042707154146627647832, −18.269033097875990316507319931584, −17.61972287425101875532373646025, −16.51922470349364623489658320724, −14.83198533873399906847331308472, −13.41643453051203828157973988185, −11.956373727740704105680429338707, −10.74074979750747559997644955886, −9.44566152389131881369715831192, −7.38790752248483336704332902315, −5.81981644686087148920681986787, −4.70920812840911269661655082592, −1.869939273182751683408212296752, 1.869939273182751683408212296752, 4.70920812840911269661655082592, 5.81981644686087148920681986787, 7.38790752248483336704332902315, 9.44566152389131881369715831192, 10.74074979750747559997644955886, 11.956373727740704105680429338707, 13.41643453051203828157973988185, 14.83198533873399906847331308472, 16.51922470349364623489658320724, 17.61972287425101875532373646025, 18.269033097875990316507319931584, 20.30638125042707154146627647832, 21.66563972555785204780870557439, 22.32883476081719799605147614760, 24.04471027020575246774989117385, 24.681483218497409008454757604732, 26.442423451163784130338143627684, 27.60237248544315232420294111024, 28.77893942445047647072544291977, 29.60681027223561542267337147912, 30.7837343166950765025407660372, 32.488723456435195652185644207532, 33.60625213311540800317874948376, 34.15346729684272106043380578223

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