Properties

Label 2-444-111.14-c1-0-10
Degree $2$
Conductor $444$
Sign $-0.114 + 0.993i$
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 + (−6.96 − 1.86i)13-s + (7.83 + 2.09i)19-s + (−7.79 − 4.5i)21-s + (4.33 + 2.5i)25-s − 5.19i·27-s + (4.63 + 4.63i)31-s + (5 − 3.46i)37-s + (−12.0 + 3.23i)39-s + (3.56 − 3.56i)43-s + (−10 + 17.3i)49-s + (13.5 − 3.63i)57-s + (1.29 − 4.83i)61-s + ⋯
L(s)  = 1  + (0.866 − 0.499i)3-s + (−0.981 − 1.70i)7-s + (0.5 − 0.866i)9-s + (−1.93 − 0.517i)13-s + (1.79 + 0.481i)19-s + (−1.70 − 0.981i)21-s + (0.866 + 0.5i)25-s − 0.999i·27-s + (0.832 + 0.832i)31-s + (0.821 − 0.569i)37-s + (−1.93 + 0.517i)39-s + (0.543 − 0.543i)43-s + (−1.42 + 2.47i)49-s + (1.79 − 0.481i)57-s + (0.165 − 0.618i)61-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.114 + 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.114 + 0.993i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (125, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.114 + 0.993i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.01718 - 1.14142i\)
\(L(\frac12)\) \(\approx\) \(1.01718 - 1.14142i\)
\(L(\frac{3}{2})\) 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 \( 1 + (6.96 + 1.86i)T + (11.2 + 6.5i)T^{2} \)
17 \( 1 + (14.7 - 8.5i)T^{2} \)
19 \( 1 + (-7.83 - 2.09i)T + (16.4 + 9.5i)T^{2} \)
23 \( 1 + 23iT^{2} \)
29 \( 1 - 29iT^{2} \)
31 \( 1 + (-4.63 - 4.63i)T + 31iT^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + (-3.56 + 3.56i)T - 43iT^{2} \)
47 \( 1 - 47T^{2} \)
53 \( 1 + (26.5 + 45.8i)T^{2} \)
59 \( 1 + (51.0 - 29.5i)T^{2} \)
61 \( 1 + (-1.29 + 4.83i)T + (-52.8 - 30.5i)T^{2} \)
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 \( 1 + (-15.1 - 4.06i)T + (68.4 + 39.5i)T^{2} \)
83 \( 1 + (41.5 + 71.8i)T^{2} \)
89 \( 1 + (-77.0 + 44.5i)T^{2} \)
97 \( 1 + (12.0 - 12.0i)T - 97iT^{2} \)
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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

−10.56980582940046249418540090389, −9.852586600787591170889488631820, −9.300406290182028320290830601860, −7.68238330122320676024597655696, −7.43822168486282783827279753220, −6.55494915967014818488398778498, −4.94033603790574622855528503646, −3.64644432560875225657879954395, −2.81447436351838292959351049710, −0.887688086153105071811725621281, 2.49354749339934097325669719097, 2.95571199940987021376400353711, 4.61556138714016531312696844683, 5.49517127755256664676024835930, 6.79986938416565620677541565237, 7.81275116303150332363630394968, 8.901311815988092499513275514805, 9.600251496929512672666684690145, 9.911566662500873046289978076543, 11.53102115725122919783813354263

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