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Properties

Label 2-44-44.43-c1-0-3
Degree $2$
Conductor $44$
Sign $0.0258 + 0.999i$
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  + (−0.707 − 1.22i)2-s − 1.73i·3-s + (−0.999 + 1.73i)4-s − 5-s + (−2.12 + 1.22i)6-s + 2.82·7-s + 2.82·8-s + (0.707 + 1.22i)10-s + (−2.82 + 1.73i)11-s + (2.99 + 1.73i)12-s + 4.89i·13-s + (−2.00 − 3.46i)14-s + 1.73i·15-s + (−2.00 − 3.46i)16-s − 4.89i·17-s + ⋯
L(s)  = 1  + (−0.499 − 0.866i)2-s − 0.999i·3-s + (−0.499 + 0.866i)4-s − 0.447·5-s + (−0.866 + 0.499i)6-s + 1.06·7-s + 0.999·8-s + (0.223 + 0.387i)10-s + (−0.852 + 0.522i)11-s + (0.866 + 0.499i)12-s + 1.35i·13-s + (−0.534 − 0.925i)14-s + 0.447i·15-s + (−0.500 − 0.866i)16-s − 1.18i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0258 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(44\)    =    \(2^{2} \cdot 11\)
Sign: $0.0258 + 0.999i$
Analytic conductor: \(0.351341\)
Root analytic conductor: \(0.592740\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{44} (43, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 44,\ (\ :1/2),\ 0.0258 + 0.999i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.450425 - 0.438922i\)
\(L(\frac12)\) \(\approx\) \(0.450425 - 0.438922i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.707 + 1.22i)T \)
11 \( 1 + (2.82 - 1.73i)T \)
good3 \( 1 + 1.73iT - 3T^{2} \)
5 \( 1 + T + 5T^{2} \)
7 \( 1 - 2.82T + 7T^{2} \)
13 \( 1 - 4.89iT - 13T^{2} \)
17 \( 1 + 4.89iT - 17T^{2} \)
19 \( 1 + 2.82T + 19T^{2} \)
23 \( 1 - 5.19iT - 23T^{2} \)
29 \( 1 - 29T^{2} \)
31 \( 1 + 1.73iT - 31T^{2} \)
37 \( 1 - 3T + 37T^{2} \)
41 \( 1 - 4.89iT - 41T^{2} \)
43 \( 1 - 5.65T + 43T^{2} \)
47 \( 1 + 3.46iT - 47T^{2} \)
53 \( 1 + 2T + 53T^{2} \)
59 \( 1 + 1.73iT - 59T^{2} \)
61 \( 1 + 9.79iT - 61T^{2} \)
67 \( 1 + 8.66iT - 67T^{2} \)
71 \( 1 - 12.1iT - 71T^{2} \)
73 \( 1 + 4.89iT - 73T^{2} \)
79 \( 1 + 11.3T + 79T^{2} \)
83 \( 1 + 83T^{2} \)
89 \( 1 + T + 89T^{2} \)
97 \( 1 - 7T + 97T^{2} \)
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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

−15.81401990658736986424387463214, −14.10886435654514023290549389578, −13.11277326131211215678804081252, −11.91898139156864932402866662584, −11.25056523370125613338279173511, −9.594068749066549720741682135531, −8.042101771015932370096598307014, −7.25570501173424324734595203834, −4.53416142277537061148088125179, −1.97651492181291732003321478328, 4.35864749049661509010047534987, 5.66334158599060577035615438446, 7.77987593831077038005442092143, 8.580813170301260414907807479267, 10.32122427851470899310251819286, 10.84085818899298643860310777797, 12.94510625872116613714527522054, 14.55825437988958034020594415929, 15.28284601673200190413807684082, 16.01477437399476310153385082590

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