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Properties

Label 2-44-44.7-c1-0-2
Degree $2$
Conductor $44$
Sign $0.708 + 0.705i$
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  + (−0.0737 − 1.41i)2-s + (1.70 + 0.554i)3-s + (−1.98 + 0.208i)4-s + (−2.39 + 1.74i)5-s + (0.656 − 2.44i)6-s + (−0.815 − 2.51i)7-s + (0.440 + 2.79i)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 + (−3.48 + 1.33i)14-s + (−5.05 + 1.64i)15-s + (3.91 − 0.828i)16-s + (0.468 + 0.644i)17-s + ⋯
L(s)  = 1  + (−0.0521 − 0.998i)2-s + (0.984 + 0.319i)3-s + (−0.994 + 0.104i)4-s + (−1.07 + 0.779i)5-s + (0.268 − 0.999i)6-s + (−0.308 − 0.948i)7-s + (0.155 + 0.987i)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 + (−0.931 + 0.357i)14-s + (−1.30 + 0.424i)15-s + (0.978 − 0.207i)16-s + (0.113 + 0.156i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.708 + 0.705i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.708 + 0.705i$
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: \((2,\ 44,\ (\ :1/2),\ 0.708 + 0.705i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.749457 - 0.309318i\)
\(L(\frac12)\) \(\approx\) \(0.749457 - 0.309318i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.0737 + 1.41i)T \)
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

−15.39530442549097393334868948324, −14.56638391136480686208676572514, −13.54227936732645760471761231135, −12.16800633594278936353688588773, −10.91760408767535257154984502741, −9.924465828474748225584054276566, −8.554776870484596482306572139424, −7.27963159009397202645305742803, −4.13958081293968062323330834959, −3.16150572257945797452774858010, 3.77428715166864530891938463609, 5.73304578858492835897105790577, 7.57628434250880428178986628082, 8.590854788380287971066836602559, 9.162463057963504090476840031582, 11.66884408491895864191720218740, 12.92563757874377212582111775879, 13.99060692433947901166954512860, 15.04525443891162860778575257120, 16.02196323579188038125225044186

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