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Properties

Label 1-44-44.39-r0-0-0
Degree $1$
Conductor $44$
Sign $0.0219 - 0.999i$
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  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s − 21-s − 23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)35-s + ⋯
L(s)  = 1  + (−0.309 − 0.951i)3-s + (−0.809 − 0.587i)5-s + (0.309 − 0.951i)7-s + (−0.809 + 0.587i)9-s + (0.809 − 0.587i)13-s + (−0.309 + 0.951i)15-s + (0.809 + 0.587i)17-s + (0.309 + 0.951i)19-s − 21-s − 23-s + (0.309 + 0.951i)25-s + (0.809 + 0.587i)27-s + (−0.309 + 0.951i)29-s + (0.809 − 0.587i)31-s + (−0.809 + 0.587i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0219 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.0219 - 0.999i$
Analytic conductor: \(0.204335\)
Root analytic conductor: \(0.204335\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (39, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 44,\ (0:\ ),\ 0.0219 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.4899008642 - 0.4792467408i\)
\(L(\frac12)\) \(\approx\) \(0.4899008642 - 0.4792467408i\)
\(L(1)\) \(\approx\) \(0.7429612369 - 0.3821233566i\)
\(L(1)\) \(\approx\) \(0.7429612369 - 0.3821233566i\)

Euler product

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

−34.407221991536566111488685644393, −33.929069024613494657079829675412, −32.383002553816703457320140701277, −31.40521155239527346661654995398, −30.32715213680286529406744366768, −28.53109794431504320973121461102, −27.79524248221390069481743745937, −26.69836235912126662643372863930, −25.64299916894155709051961426124, −23.90193374785026271025819591919, −22.74264966217573713109095467762, −21.7542422121599441589396765521, −20.61243456963227716510299464494, −19.067952204636833342761542834375, −17.882823231692953432243719961538, −16.150547964773457396473462299288, −15.41264326847589838028873256195, −14.208003256942148246122632181874, −11.9126332890027883379863664126, −11.20111513440706741368472808586, −9.6122984973181065380995033445, −8.189588180294514304200876334714, −6.22157163339266068768141328544, −4.61598405018775830388225912214, −3.07457062346429072077383180313, 1.20270054813225589025901463549, 3.82825627270565497325032183999, 5.69986468850286818448552360040, 7.46769421635951922036780657561, 8.29864635473136537466594232893, 10.59239217892703775245217535174, 11.90898968161931291877373286147, 12.99528605282832326058096411844, 14.29405173762946298118078071244, 16.14269374988042029604020903963, 17.19595475749757764929746090536, 18.52110868805147425187344992443, 19.76702176492772790466513659689, 20.7043057378583660392381210804, 22.80279818559651353811869495500, 23.56735060684927979139993821679, 24.46678514851704878841018695538, 25.84213043395276834001115029733, 27.40750212186395108415166934016, 28.32777995165993932285366141009, 29.75057964828349764906048153175, 30.53916856002816048445495062229, 31.72429478816138063461437609706, 33.09715077502467122621464654378, 34.429087586134657245546699426965

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