Properties

Label 2-444-111.107-c0-0-0
Degree $2$
Conductor $444$
Sign $0.0357 - 0.999i$
Analytic cond. $0.221584$
Root an. cond. $0.470728$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (0.173 + 0.984i)3-s + (−1.43 + 1.20i)7-s + (−0.939 + 0.342i)9-s + (1.76 + 0.642i)13-s + (−0.173 − 0.984i)19-s + (−1.43 − 1.20i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + 1.53·31-s + (−0.5 + 0.866i)37-s + (−0.326 + 1.85i)39-s + 0.347·43-s + (0.439 − 2.49i)49-s + (0.939 − 0.342i)57-s + (−1.87 − 0.684i)61-s + ⋯
L(s)  = 1  + (0.173 + 0.984i)3-s + (−1.43 + 1.20i)7-s + (−0.939 + 0.342i)9-s + (1.76 + 0.642i)13-s + (−0.173 − 0.984i)19-s + (−1.43 − 1.20i)21-s + (0.173 − 0.984i)25-s + (−0.5 − 0.866i)27-s + 1.53·31-s + (−0.5 + 0.866i)37-s + (−0.326 + 1.85i)39-s + 0.347·43-s + (0.439 − 2.49i)49-s + (0.939 − 0.342i)57-s + (−1.87 − 0.684i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0357 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0357 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.0357 - 0.999i$
Analytic conductor: \(0.221584\)
Root analytic conductor: \(0.470728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (329, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :0),\ 0.0357 - 0.999i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7946202244\)
\(L(\frac12)\) \(\approx\) \(0.7946202244\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-0.173 - 0.984i)T \)
37 \( 1 + (0.5 - 0.866i)T \)
good5 \( 1 + (-0.173 + 0.984i)T^{2} \)
7 \( 1 + (1.43 - 1.20i)T + (0.173 - 0.984i)T^{2} \)
11 \( 1 + (0.5 + 0.866i)T^{2} \)
13 \( 1 + (-1.76 - 0.642i)T + (0.766 + 0.642i)T^{2} \)
17 \( 1 + (-0.766 + 0.642i)T^{2} \)
19 \( 1 + (0.173 + 0.984i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (0.5 - 0.866i)T^{2} \)
29 \( 1 + (0.5 + 0.866i)T^{2} \)
31 \( 1 - 1.53T + T^{2} \)
41 \( 1 + (-0.766 - 0.642i)T^{2} \)
43 \( 1 - 0.347T + T^{2} \)
47 \( 1 + (0.5 - 0.866i)T^{2} \)
53 \( 1 + (-0.173 - 0.984i)T^{2} \)
59 \( 1 + (-0.173 - 0.984i)T^{2} \)
61 \( 1 + (1.87 + 0.684i)T + (0.766 + 0.642i)T^{2} \)
67 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
71 \( 1 + (0.939 - 0.342i)T^{2} \)
73 \( 1 - 0.347T + T^{2} \)
79 \( 1 + (-0.266 + 0.223i)T + (0.173 - 0.984i)T^{2} \)
83 \( 1 + (-0.766 + 0.642i)T^{2} \)
89 \( 1 + (-0.173 - 0.984i)T^{2} \)
97 \( 1 + (0.173 + 0.300i)T + (-0.5 + 0.866i)T^{2} \)
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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

−11.47957532648526403131036122429, −10.54662095436613263115244780560, −9.644762853468066933721844734192, −8.920061490094726554956795937150, −8.394592117693145323950168448402, −6.48900471738461139131922629117, −6.05798112110947751284563553068, −4.73399647459845896232903938942, −3.53903613353404803245005779831, −2.63526788444988798891296065348, 1.15292438492964800010818308943, 3.11452213745379758835487578601, 3.86244551414909623103377178121, 5.85060427399910958311401958953, 6.45398735054374128712646973347, 7.36665263238220499019184201563, 8.246231445867409386508758485141, 9.243877950904694662304771952960, 10.33548733127371334356855104450, 11.00534592247317498848856675513

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