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Properties

Label 2-44-11.4-c1-0-0
Degree $2$
Conductor $44$
Sign $0.751 - 0.659i$
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  + (−0.809 + 2.48i)3-s + (1.30 − 0.951i)5-s + (−1.19 − 3.66i)7-s + (−3.11 − 2.26i)9-s + (1.23 + 3.07i)11-s + (−1.92 − 1.40i)13-s + (1.30 + 4.02i)15-s + (−1.92 + 1.40i)17-s + (1.19 − 3.66i)19-s + 10.0·21-s − 2.47·23-s + (−0.736 + 2.26i)25-s + (1.80 − 1.31i)27-s + (2.66 + 8.19i)29-s + (−0.690 − 0.502i)31-s + ⋯
L(s)  = 1  + (−0.467 + 1.43i)3-s + (0.585 − 0.425i)5-s + (−0.450 − 1.38i)7-s + (−1.03 − 0.755i)9-s + (0.372 + 0.927i)11-s + (−0.534 − 0.388i)13-s + (0.337 + 1.04i)15-s + (−0.467 + 0.339i)17-s + (0.273 − 0.840i)19-s + 2.20·21-s − 0.515·23-s + (−0.147 + 0.453i)25-s + (0.348 − 0.252i)27-s + (0.494 + 1.52i)29-s + (−0.124 − 0.0901i)31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.751 - 0.659i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.751 - 0.659i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (37, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.751 - 0.659i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.685708 + 0.258039i\)
\(L(\frac12)\) \(\approx\) \(0.685708 + 0.258039i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-1.23 - 3.07i)T \)
good3 \( 1 + (0.809 - 2.48i)T + (-2.42 - 1.76i)T^{2} \)
5 \( 1 + (-1.30 + 0.951i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (1.19 + 3.66i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (1.92 + 1.40i)T + (4.01 + 12.3i)T^{2} \)
17 \( 1 + (1.92 - 1.40i)T + (5.25 - 16.1i)T^{2} \)
19 \( 1 + (-1.19 + 3.66i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 2.47T + 23T^{2} \)
29 \( 1 + (-2.66 - 8.19i)T + (-23.4 + 17.0i)T^{2} \)
31 \( 1 + (0.690 + 0.502i)T + (9.57 + 29.4i)T^{2} \)
37 \( 1 + (0.572 + 1.76i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-2.66 + 8.19i)T + (-33.1 - 24.0i)T^{2} \)
43 \( 1 + 43T^{2} \)
47 \( 1 + (-0.427 + 1.31i)T + (-38.0 - 27.6i)T^{2} \)
53 \( 1 + (-3.30 - 2.40i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-0.336 - 1.03i)T + (-47.7 + 34.6i)T^{2} \)
61 \( 1 + (1.92 - 1.40i)T + (18.8 - 58.0i)T^{2} \)
67 \( 1 - 12.9T + 67T^{2} \)
71 \( 1 + (-5.16 + 3.75i)T + (21.9 - 67.5i)T^{2} \)
73 \( 1 + (0.281 + 0.865i)T + (-59.0 + 42.9i)T^{2} \)
79 \( 1 + (-5.78 - 4.20i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (10.5 - 7.66i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 - 0.472T + 89T^{2} \)
97 \( 1 + (-11.7 - 8.55i)T + (29.9 + 92.2i)T^{2} \)
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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

−16.19215577536246413583005949780, −15.14444277766895215778844758659, −13.87744701164209751287805826993, −12.59777131567517146311068589982, −10.90878761946003941956126427944, −10.07284856937093834242920925634, −9.241595476930417866938762169999, −7.03567637021931389353393387013, −5.17794652531498860175810295267, −3.99409198562467247986882310945, 2.38029506608655636628223600595, 5.85069673653490757538201885893, 6.51488911598948432479578009822, 8.178532551950523448280781131337, 9.664287876344252573961175056223, 11.54611805486196835656903355372, 12.23148951211053315905871959640, 13.40522188650574785940462430679, 14.36820429288925606211253232519, 15.91983111659509013433208729680

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