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Properties

Label 2-44-44.7-c1-0-1
Degree $2$
Conductor $44$
Sign $0.600 - 0.799i$
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  + (0.544 + 1.30i)2-s + (−0.704 − 0.228i)3-s + (−1.40 + 1.42i)4-s + (1.09 − 0.792i)5-s + (−0.0847 − 1.04i)6-s + (−0.503 − 1.54i)7-s + (−2.62 − 1.06i)8-s + (−1.98 − 1.44i)9-s + (1.62 + 0.992i)10-s + (3.29 + 0.357i)11-s + (1.31 − 0.678i)12-s + (−2.09 + 2.87i)13-s + (1.74 − 1.49i)14-s + (−0.949 + 0.308i)15-s + (−0.0391 − 3.99i)16-s + (−0.0411 − 0.0566i)17-s + ⋯
L(s)  = 1  + (0.384 + 0.922i)2-s + (−0.406 − 0.132i)3-s + (−0.703 + 0.710i)4-s + (0.487 − 0.354i)5-s + (−0.0345 − 0.426i)6-s + (−0.190 − 0.585i)7-s + (−0.926 − 0.375i)8-s + (−0.661 − 0.480i)9-s + (0.514 + 0.313i)10-s + (0.994 + 0.107i)11-s + (0.380 − 0.196i)12-s + (−0.579 + 0.798i)13-s + (0.467 − 0.400i)14-s + (−0.245 + 0.0796i)15-s + (−0.00977 − 0.999i)16-s + (−0.00997 − 0.0137i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.600 - 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.600 - 0.799i$
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: \((2,\ 44,\ (\ :1/2),\ 0.600 - 0.799i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.733525 + 0.366530i\)
\(L(\frac12)\) \(\approx\) \(0.733525 + 0.366530i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (-0.544 - 1.30i)T \)
11 \( 1 + (-3.29 - 0.357i)T \)
good3 \( 1 + (0.704 + 0.228i)T + (2.42 + 1.76i)T^{2} \)
5 \( 1 + (-1.09 + 0.792i)T + (1.54 - 4.75i)T^{2} \)
7 \( 1 + (0.503 + 1.54i)T + (-5.66 + 4.11i)T^{2} \)
13 \( 1 + (2.09 - 2.87i)T + (-4.01 - 12.3i)T^{2} \)
17 \( 1 + (0.0411 + 0.0566i)T + (-5.25 + 16.1i)T^{2} \)
19 \( 1 + (2.45 - 7.56i)T + (-15.3 - 11.1i)T^{2} \)
23 \( 1 + 5.30iT - 23T^{2} \)
29 \( 1 + (-3.74 + 1.21i)T + (23.4 - 17.0i)T^{2} \)
31 \( 1 + (-2.17 + 2.98i)T + (-9.57 - 29.4i)T^{2} \)
37 \( 1 + (2.12 + 6.54i)T + (-29.9 + 21.7i)T^{2} \)
41 \( 1 + (-5.50 - 1.78i)T + (33.1 + 24.0i)T^{2} \)
43 \( 1 + 1.89T + 43T^{2} \)
47 \( 1 + (9.32 + 3.03i)T + (38.0 + 27.6i)T^{2} \)
53 \( 1 + (-3.14 - 2.28i)T + (16.3 + 50.4i)T^{2} \)
59 \( 1 + (-4.62 + 1.50i)T + (47.7 - 34.6i)T^{2} \)
61 \( 1 + (-0.791 - 1.08i)T + (-18.8 + 58.0i)T^{2} \)
67 \( 1 - 4.91iT - 67T^{2} \)
71 \( 1 + (-4.10 - 5.64i)T + (-21.9 + 67.5i)T^{2} \)
73 \( 1 + (9.62 - 3.12i)T + (59.0 - 42.9i)T^{2} \)
79 \( 1 + (4.85 + 3.52i)T + (24.4 + 75.1i)T^{2} \)
83 \( 1 + (2.92 - 2.12i)T + (25.6 - 78.9i)T^{2} \)
89 \( 1 + 5.39T + 89T^{2} \)
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

−16.53568433847766940527431426265, −14.69645711319985510099915383135, −14.12654511497398768764337215755, −12.72679619320294847880869774966, −11.78894029883818396842844906406, −9.802112041520851902368750849626, −8.581648436867149380485173759237, −6.87355935106800023398815057217, −5.86096355162820785618342712053, −4.12555725224630642994296772726, 2.76082777466018957972996401558, 4.98840083919941029063903722522, 6.28526000681873083224300132354, 8.760102592212026320290881488395, 9.993043702724991250188696460406, 11.15487345637810827976489733035, 12.07727684784166784335917958057, 13.38766290204612536592493642523, 14.39682276708884974690743027416, 15.52735994686192182852703607386

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