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Properties

Label 2-444-111.23-c1-0-6
Degree $2$
Conductor $444$
Sign $0.998 - 0.0501i$
Analytic cond. $3.54535$
Root an. cond. $1.88291$
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  + (1.5 − 0.866i)3-s + (2.59 + 4.5i)7-s + (1.5 − 2.59i)9-s + (−0.0358 + 0.133i)13-s + (−0.830 + 3.09i)19-s + (7.79 + 4.5i)21-s + (−4.33 − 2.5i)25-s − 5.19i·27-s + (6.36 − 6.36i)31-s + (5 − 3.46i)37-s + (0.0621 + 0.232i)39-s + (−8.56 − 8.56i)43-s + (−10 + 17.3i)49-s + (1.43 + 5.36i)57-s + (−14.2 − 3.83i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (0.981 + 1.70i)7-s + (0.5 − 0.866i)9-s + (−0.00995 + 0.0371i)13-s + (−0.190 + 0.710i)19-s + (1.70 + 0.981i)21-s + (−0.866 − 0.5i)25-s − 0.999i·27-s + (1.14 − 1.14i)31-s + (0.821 − 0.569i)37-s + (0.00995 + 0.0371i)39-s + (−1.30 − 1.30i)43-s + (−1.42 + 2.47i)49-s + (0.190 + 0.710i)57-s + (−1.83 − 0.490i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0501i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 - 0.0501i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.998 - 0.0501i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (245, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.998 - 0.0501i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.00466 + 0.0502970i\)
\(L(\frac12)\) \(\approx\) \(2.00466 + 0.0502970i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 + (-1.5 + 0.866i)T \)
37 \( 1 + (-5 + 3.46i)T \)
good5 \( 1 + (4.33 + 2.5i)T^{2} \)
7 \( 1 + (-2.59 - 4.5i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 + 11T^{2} \)
13 \( 1 + (0.0358 - 0.133i)T + (-11.2 - 6.5i)T^{2} \)
17 \( 1 + (-14.7 + 8.5i)T^{2} \)
19 \( 1 + (0.830 - 3.09i)T + (-16.4 - 9.5i)T^{2} \)
23 \( 1 - 23iT^{2} \)
29 \( 1 + 29iT^{2} \)
31 \( 1 + (-6.36 + 6.36i)T - 31iT^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (8.56 + 8.56i)T + 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (-51.0 + 29.5i)T^{2} \)
61 \( 1 + (14.2 + 3.83i)T + (52.8 + 30.5i)T^{2} \)
67 \( 1 + (-4.33 + 2.5i)T + (33.5 - 58.0i)T^{2} \)
71 \( 1 + (35.5 - 61.4i)T^{2} \)
73 \( 1 - 17iT - 73T^{2} \)
79 \( 1 + (2.16 - 8.06i)T + (-68.4 - 39.5i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (77.0 - 44.5i)T^{2} \)
97 \( 1 + (6.90 + 6.90i)T + 97iT^{2} \)
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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

−11.41769989513578869643706112697, −9.982537222008737534609226652197, −9.124562941591658578849268224420, −8.290723799781407093760720252680, −7.83177935415582389992535979139, −6.40089908570384358654564358401, −5.51317404320217304683832855196, −4.18194976324298066619874275735, −2.68023172231866565934671332033, −1.81890710963619590499439279143, 1.48635667794053151497638489993, 3.15088367685776090174303619579, 4.29949189951232798577420508262, 4.89848701180947344926063676420, 6.69670621028378898455155362507, 7.68417107390136938149518762026, 8.206655838172893104963125937298, 9.368213070418357287519897994432, 10.27571188425527282388698082214, 10.85283724201053514863457982440

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