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Properties

Label 2-44-11.9-c1-0-0
Degree $2$
Conductor $44$
Sign $0.998 + 0.0475i$
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  + (0.309 − 0.224i)3-s + (0.190 + 0.587i)5-s + (−2.30 − 1.67i)7-s + (−0.881 + 2.71i)9-s + (−3.23 + 0.726i)11-s + (1.42 − 4.39i)13-s + (0.190 + 0.138i)15-s + (1.42 + 4.39i)17-s + (2.30 − 1.67i)19-s − 1.09·21-s + 6.47·23-s + (3.73 − 2.71i)25-s + (0.690 + 2.12i)27-s + (−5.16 − 3.75i)29-s + (−1.80 + 5.56i)31-s + ⋯
L(s)  = 1  + (0.178 − 0.129i)3-s + (0.0854 + 0.262i)5-s + (−0.872 − 0.634i)7-s + (−0.293 + 0.904i)9-s + (−0.975 + 0.219i)11-s + (0.395 − 1.21i)13-s + (0.0493 + 0.0358i)15-s + (0.346 + 1.06i)17-s + (0.529 − 0.384i)19-s − 0.237·21-s + 1.34·23-s + (0.747 − 0.542i)25-s + (0.132 + 0.409i)27-s + (−0.958 − 0.696i)29-s + (−0.324 + 0.999i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0475i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.998 + 0.0475i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (9, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.998 + 0.0475i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.787116 - 0.0187163i\)
\(L(\frac12)\) \(\approx\) \(0.787116 - 0.0187163i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
11 \( 1 + (3.23 - 0.726i)T \)
good3 \( 1 + (-0.309 + 0.224i)T + (0.927 - 2.85i)T^{2} \)
5 \( 1 + (-0.190 - 0.587i)T + (-4.04 + 2.93i)T^{2} \)
7 \( 1 + (2.30 + 1.67i)T + (2.16 + 6.65i)T^{2} \)
13 \( 1 + (-1.42 + 4.39i)T + (-10.5 - 7.64i)T^{2} \)
17 \( 1 + (-1.42 - 4.39i)T + (-13.7 + 9.99i)T^{2} \)
19 \( 1 + (-2.30 + 1.67i)T + (5.87 - 18.0i)T^{2} \)
23 \( 1 - 6.47T + 23T^{2} \)
29 \( 1 + (5.16 + 3.75i)T + (8.96 + 27.5i)T^{2} \)
31 \( 1 + (1.80 - 5.56i)T + (-25.0 - 18.2i)T^{2} \)
37 \( 1 + (3.92 + 2.85i)T + (11.4 + 35.1i)T^{2} \)
41 \( 1 + (5.16 - 3.75i)T + (12.6 - 38.9i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (2.92 - 2.12i)T + (14.5 - 44.6i)T^{2} \)
53 \( 1 + (-2.19 + 6.74i)T + (-42.8 - 31.1i)T^{2} \)
59 \( 1 + (-8.16 - 5.93i)T + (18.2 + 56.1i)T^{2} \)
61 \( 1 + (-1.42 - 4.39i)T + (-49.3 + 35.8i)T^{2} \)
67 \( 1 + 4.94T + 67T^{2} \)
71 \( 1 + (2.66 + 8.19i)T + (-57.4 + 41.7i)T^{2} \)
73 \( 1 + (-9.78 - 7.10i)T + (22.5 + 69.4i)T^{2} \)
79 \( 1 + (4.28 - 13.1i)T + (-63.9 - 46.4i)T^{2} \)
83 \( 1 + (4.95 + 15.2i)T + (-67.1 + 48.7i)T^{2} \)
89 \( 1 + 8.47T + 89T^{2} \)
97 \( 1 + (-1.71 + 5.29i)T + (-78.4 - 57.0i)T^{2} \)
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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

−15.96149791675785783370759128850, −14.81362285379978402131969981782, −13.35997225794383601822631884748, −12.87446740415669975610196552064, −10.86136025301780892870358543897, −10.16218333175933919274102854969, −8.351790534203467314088545874729, −7.12529478523592729398288261402, −5.39087715856145334109892895116, −3.11308547414497674701489811645, 3.20960585794750891256027177047, 5.42139156431851097982450816862, 6.94480028430097530891688252187, 8.860722555420078874581113932709, 9.588069203806900765420371235017, 11.32773239014164160560229472553, 12.49690927024069757868065442421, 13.59152987711081956716680886640, 14.91800591630508743391222056174, 15.96921180886278814725872961618

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