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Properties

Label 2-44-44.7-c1-0-3
Degree $2$
Conductor $44$
Sign $0.934 + 0.355i$
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  + (1.36 − 0.366i)2-s + (−1.70 − 0.554i)3-s + (1.73 − 1.00i)4-s + (−2.39 + 1.74i)5-s + (−2.53 − 0.132i)6-s + (0.815 + 2.51i)7-s + (1.99 − 2.00i)8-s + (0.174 + 0.126i)9-s + (−2.63 + 3.26i)10-s + (−1.40 − 3.00i)11-s + (−3.50 + 0.747i)12-s + (1.39 − 1.92i)13-s + (2.03 + 3.13i)14-s + (5.05 − 1.64i)15-s + (1.99 − 3.46i)16-s + (0.468 + 0.644i)17-s + ⋯
L(s)  = 1  + (0.965 − 0.259i)2-s + (−0.984 − 0.319i)3-s + (0.865 − 0.500i)4-s + (−1.07 + 0.779i)5-s + (−1.03 − 0.0539i)6-s + (0.308 + 0.948i)7-s + (0.706 − 0.707i)8-s + (0.0580 + 0.0421i)9-s + (−0.834 + 1.03i)10-s + (−0.425 − 0.905i)11-s + (−1.01 + 0.215i)12-s + (0.388 − 0.534i)13-s + (0.543 + 0.836i)14-s + (1.30 − 0.424i)15-s + (0.499 − 0.866i)16-s + (0.113 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.934 + 0.355i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.934 + 0.355i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (7, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.934 + 0.355i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.907889 - 0.166659i\)
\(L(\frac12)\) \(\approx\) \(0.907889 - 0.166659i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-1.36 + 0.366i)T \)
11 \( 1 + (1.40 + 3.00i)T \)
good3 \( 1 + (1.70 + 0.554i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (2.39 - 1.74i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (-0.815 - 2.51i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (-1.39 + 1.92i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (-0.468 - 0.644i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (0.624 - 1.92i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 - 2.61iT - 23T^{2} \)
29 \( 1 + (1.08 - 0.351i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-3.51 + 4.84i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (-2.69 - 8.30i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (9.28 + 3.01i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 6.14T + 43T^{2} \)
47 \( 1 + (3.31 + 1.07i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-6.63 - 4.82i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (0.712 - 0.231i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (3.71 + 5.11i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 + 5.40iT - 67T^{2} \)
71 \( 1 + (-2.56 - 3.53i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (-0.845 + 0.274i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (-2.17 - 1.58i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (-12.1 + 8.85i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 4.25T + 89T^{2} \)
97 \( 1 + (5.92 + 4.30i)T + (29.9 + 92.2i)T^{2} \)
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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

−15.52065305105167513226197453227, −14.94102641436383813536834837757, −13.40522098115577928413217851045, −11.99850040859835058959925269508, −11.54698823036479209096413876670, −10.59674308782341090692694324536, −8.049616327185247093804039339896, −6.44039295670537873745539656767, −5.38726880851750830540119245571, −3.29789081753004457615719220340, 4.21844425640318850237605637302, 5.02909577387759671992870111748, 6.88065201305950286686064102684, 8.166427226356224603415539164854, 10.55316959368569257632896860251, 11.54528231449592106268446951384, 12.37441742274047673044304721838, 13.62129907683157087299827304335, 15.02851644067673355183612157574, 16.16048181475930077477007359129

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