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Properties

Label 2-444-37.10-c1-0-1
Degree $2$
Conductor $444$
Sign $0.729 - 0.683i$
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  + (0.5 + 0.866i)3-s + (−1 − 1.73i)5-s + (1.5 + 2.59i)7-s + (−0.499 + 0.866i)9-s + 2·11-s + (1.5 + 2.59i)13-s + (0.999 − 1.73i)15-s + (2 + 3.46i)19-s + (−1.5 + 2.59i)21-s + 6·23-s + (0.500 − 0.866i)25-s − 0.999·27-s + 31-s + (1 + 1.73i)33-s + (3 − 5.19i)35-s + ⋯
L(s)  = 1  + (0.288 + 0.499i)3-s + (−0.447 − 0.774i)5-s + (0.566 + 0.981i)7-s + (−0.166 + 0.288i)9-s + 0.603·11-s + (0.416 + 0.720i)13-s + (0.258 − 0.447i)15-s + (0.458 + 0.794i)19-s + (−0.327 + 0.566i)21-s + 1.25·23-s + (0.100 − 0.173i)25-s − 0.192·27-s + 0.179·31-s + (0.174 + 0.301i)33-s + (0.507 − 0.878i)35-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.729 - 0.683i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.729 - 0.683i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (121, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.729 - 0.683i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.44703 + 0.572094i\)
\(L(\frac12)\) \(\approx\) \(1.44703 + 0.572094i\)
\(L(\frac{3}{2})\) 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 \( 1 + (-0.5 - 0.866i)T \)
37 \( 1 + (5 - 3.46i)T \)
good5 \( 1 + (1 + 1.73i)T + (-2.5 + 4.33i)T^{2} \)
7 \( 1 + (-1.5 - 2.59i)T + (-3.5 + 6.06i)T^{2} \)
11 \( 1 - 2T + 11T^{2} \)
13 \( 1 + (-1.5 - 2.59i)T + (-6.5 + 11.2i)T^{2} \)
17 \( 1 + (-8.5 - 14.7i)T^{2} \)
19 \( 1 + (-2 - 3.46i)T + (-9.5 + 16.4i)T^{2} \)
23 \( 1 - 6T + 23T^{2} \)
29 \( 1 + 29T^{2} \)
31 \( 1 - T + 31T^{2} \)
41 \( 1 + (-20.5 + 35.5i)T^{2} \)
43 \( 1 + 5T + 43T^{2} \)
47 \( 1 + 4T + 47T^{2} \)
53 \( 1 + (3 - 5.19i)T + (-26.5 - 45.8i)T^{2} \)
59 \( 1 + (-5 + 8.66i)T + (-29.5 - 51.0i)T^{2} \)
61 \( 1 + (-1 - 1.73i)T + (-30.5 + 52.8i)T^{2} \)
67 \( 1 + (1.5 + 2.59i)T + (-33.5 + 58.0i)T^{2} \)
71 \( 1 + (6 + 10.3i)T + (-35.5 + 61.4i)T^{2} \)
73 \( 1 - T + 73T^{2} \)
79 \( 1 + (7.5 + 12.9i)T + (-39.5 + 68.4i)T^{2} \)
83 \( 1 + (-2 + 3.46i)T + (-41.5 - 71.8i)T^{2} \)
89 \( 1 + (-5 + 8.66i)T + (-44.5 - 77.0i)T^{2} \)
97 \( 1 - 9T + 97T^{2} \)
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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

−11.46805202967614883185329976844, −10.25281117079255679833260001538, −9.077928829848661357077135847031, −8.755504874835766274827723465077, −7.84100256421410073366864533714, −6.47773006999013369264843238739, −5.25537786487820421836300289433, −4.49452356591973116470284841820, −3.29465274599078855995045033718, −1.63708486186948917741234807068, 1.13935452837965715224780824914, 2.94192257059525144358612715118, 3.89888629739597597281491487362, 5.23433690812791219986415704380, 6.74402502505723125181065531173, 7.20355346464882933746491559319, 8.111593584125195773489752718423, 9.078887597081834615578111950667, 10.30765525895044531320704396421, 11.08986569567919564547139860628

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