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Properties

Label 2-444-111.110-c0-0-1
Degree $2$
Conductor $444$
Sign $0.5 + 0.866i$
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  + (−0.5 − 0.866i)3-s + 7-s + (−0.499 + 0.866i)9-s − 1.73i·11-s + (−0.5 − 0.866i)21-s − 25-s + 0.999·27-s + (−1.49 + 0.866i)33-s + 37-s + 1.73i·41-s + 1.73i·47-s + 1.73i·53-s + (−0.499 + 0.866i)63-s − 2·67-s − 1.73i·71-s + ⋯
L(s)  = 1  + (−0.5 − 0.866i)3-s + 7-s + (−0.499 + 0.866i)9-s − 1.73i·11-s + (−0.5 − 0.866i)21-s − 25-s + 0.999·27-s + (−1.49 + 0.866i)33-s + 37-s + 1.73i·41-s + 1.73i·47-s + 1.73i·53-s + (−0.499 + 0.866i)63-s − 2·67-s − 1.73i·71-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.5 + 0.866i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.5 + 0.866i$
Analytic conductor: \(0.221584\)
Root analytic conductor: \(0.470728\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (221, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :0),\ 0.5 + 0.866i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7824815740\)
\(L(\frac12)\) \(\approx\) \(0.7824815740\)
\(L(1)\) 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 \( 1 + (0.5 + 0.866i)T \)
37 \( 1 - T \)
good5 \( 1 + T^{2} \)
7 \( 1 - T + T^{2} \)
11 \( 1 + 1.73iT - T^{2} \)
13 \( 1 - T^{2} \)
17 \( 1 + T^{2} \)
19 \( 1 - T^{2} \)
23 \( 1 + T^{2} \)
29 \( 1 + T^{2} \)
31 \( 1 - T^{2} \)
41 \( 1 - 1.73iT - T^{2} \)
43 \( 1 - T^{2} \)
47 \( 1 - 1.73iT - T^{2} \)
53 \( 1 - 1.73iT - T^{2} \)
59 \( 1 + T^{2} \)
61 \( 1 - T^{2} \)
67 \( 1 + 2T + T^{2} \)
71 \( 1 + 1.73iT - T^{2} \)
73 \( 1 - T + T^{2} \)
79 \( 1 - T^{2} \)
83 \( 1 - 1.73iT - T^{2} \)
89 \( 1 + T^{2} \)
97 \( 1 - T^{2} \)
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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

−11.26303786876580160644958091232, −10.70417466376122999834984814659, −9.232353427821066858697442828666, −8.108327758868449584816010934364, −7.76245458016475291345774216912, −6.31590810636331028396689979234, −5.71593979959883848179323217908, −4.53613856377482178644515444175, −2.88788319861533065586617106199, −1.32356133963563538771524665473, 2.04959209147957484008262775619, 3.89359063532285634935367099334, 4.73319749530810311706120149747, 5.53460952973646042990621797065, 6.84346789363530881075378952314, 7.82908736270929421247710671950, 8.940330081524941427471309593592, 9.870505506186569477634988566105, 10.46869270846740983182702188531, 11.54844875575073403571597113739

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