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Properties

Label 2-444-12.11-c1-0-52
Degree $2$
Conductor $444$
Sign $-0.941 + 0.338i$
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  + (−1.40 + 0.163i)2-s + (−1.72 + 0.196i)3-s + (1.94 − 0.458i)4-s − 3.43i·5-s + (2.38 − 0.556i)6-s − 4.57i·7-s + (−2.65 + 0.962i)8-s + (2.92 − 0.676i)9-s + (0.560 + 4.82i)10-s + 3.04·11-s + (−3.25 + 1.17i)12-s − 6.03·13-s + (0.746 + 6.42i)14-s + (0.675 + 5.91i)15-s + (3.57 − 1.78i)16-s + 0.831i·17-s + ⋯
L(s)  = 1  + (−0.993 + 0.115i)2-s + (−0.993 + 0.113i)3-s + (0.973 − 0.229i)4-s − 1.53i·5-s + (0.973 − 0.227i)6-s − 1.72i·7-s + (−0.940 + 0.340i)8-s + (0.974 − 0.225i)9-s + (0.177 + 1.52i)10-s + 0.916·11-s + (−0.941 + 0.338i)12-s − 1.67·13-s + (0.199 + 1.71i)14-s + (0.174 + 1.52i)15-s + (0.894 − 0.446i)16-s + 0.201i·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 + 0.338i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 + 0.338i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.941 + 0.338i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (371, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.941 + 0.338i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.0765406 - 0.439198i\)
\(L(\frac12)\) \(\approx\) \(0.0765406 - 0.439198i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (1.40 - 0.163i)T \)
3 \( 1 + (1.72 - 0.196i)T \)
37 \( 1 - T \)
good5 \( 1 + 3.43iT - 5T^{2} \)
7 \( 1 + 4.57iT - 7T^{2} \)
11 \( 1 - 3.04T + 11T^{2} \)
13 \( 1 + 6.03T + 13T^{2} \)
17 \( 1 - 0.831iT - 17T^{2} \)
19 \( 1 - 2.65iT - 19T^{2} \)
23 \( 1 - 1.78T + 23T^{2} \)
29 \( 1 + 7.42iT - 29T^{2} \)
31 \( 1 - 1.01iT - 31T^{2} \)
41 \( 1 - 5.34iT - 41T^{2} \)
43 \( 1 + 8.74iT - 43T^{2} \)
47 \( 1 - 1.38T + 47T^{2} \)
53 \( 1 - 6.20iT - 53T^{2} \)
59 \( 1 + 8.65T + 59T^{2} \)
61 \( 1 + 6.78T + 61T^{2} \)
67 \( 1 - 10.8iT - 67T^{2} \)
71 \( 1 - 1.12T + 71T^{2} \)
73 \( 1 - 5.61T + 73T^{2} \)
79 \( 1 + 4.16iT - 79T^{2} \)
83 \( 1 + 1.49T + 83T^{2} \)
89 \( 1 + 15.3iT - 89T^{2} \)
97 \( 1 - 16.7T + 97T^{2} \)
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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

−10.45025911495538201270587184043, −9.872696165610458278593573800801, −9.129283617008750823464013566693, −7.81697185508912012321198768227, −7.20063198246999278599277222718, −6.13568952880479267154360026827, −4.90579620392977159479827691930, −4.08618229531363524990993748524, −1.45780413687853266663346659011, −0.45004571313629495368426931110, 2.09560038539950876261000866021, 3.04538650473874917574060299489, 5.12272403705784851893985944521, 6.26382905239393876138717036354, 6.83537584318895461210193921545, 7.65106077440420068677938005829, 9.126776530604953774668176212246, 9.697143003162157411794812758005, 10.72364470007080285060751679153, 11.38205195862937902379048342798

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