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Properties

Label 2-444-37.25-c1-0-0
Degree $2$
Conductor $444$
Sign $-0.999 - 0.0200i$
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  + (−0.766 − 0.642i)3-s + (1.08 + 2.98i)5-s + (−3.88 + 1.41i)7-s + (0.173 + 0.984i)9-s + (−2.98 − 5.17i)11-s + (−4.72 − 0.833i)13-s + (1.08 − 2.98i)15-s + (−2.47 + 0.435i)17-s + (3.57 − 4.25i)19-s + (3.88 + 1.41i)21-s + (−3.14 − 1.81i)23-s + (−3.90 + 3.27i)25-s + (0.500 − 0.866i)27-s + (−1.48 + 0.855i)29-s + 7.53i·31-s + ⋯
L(s)  = 1  + (−0.442 − 0.371i)3-s + (0.486 + 1.33i)5-s + (−1.46 + 0.534i)7-s + (0.0578 + 0.328i)9-s + (−0.901 − 1.56i)11-s + (−1.31 − 0.231i)13-s + (0.280 − 0.771i)15-s + (−0.599 + 0.105i)17-s + (0.819 − 0.976i)19-s + (0.847 + 0.308i)21-s + (−0.655 − 0.378i)23-s + (−0.781 + 0.655i)25-s + (0.0962 − 0.166i)27-s + (−0.275 + 0.158i)29-s + 1.35i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0200i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0200i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $-0.999 - 0.0200i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (25, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ -0.999 - 0.0200i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.00127502 + 0.127319i\)
\(L(\frac12)\) \(\approx\) \(0.00127502 + 0.127319i\)
\(L(\frac{3}{2})\) 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 \( 1 + (0.766 + 0.642i)T \)
37 \( 1 + (0.306 - 6.07i)T \)
good5 \( 1 + (-1.08 - 2.98i)T + (-3.83 + 3.21i)T^{2} \)
7 \( 1 + (3.88 - 1.41i)T + (5.36 - 4.49i)T^{2} \)
11 \( 1 + (2.98 + 5.17i)T + (-5.5 + 9.52i)T^{2} \)
13 \( 1 + (4.72 + 0.833i)T + (12.2 + 4.44i)T^{2} \)
17 \( 1 + (2.47 - 0.435i)T + (15.9 - 5.81i)T^{2} \)
19 \( 1 + (-3.57 + 4.25i)T + (-3.29 - 18.7i)T^{2} \)
23 \( 1 + (3.14 + 1.81i)T + (11.5 + 19.9i)T^{2} \)
29 \( 1 + (1.48 - 0.855i)T + (14.5 - 25.1i)T^{2} \)
31 \( 1 - 7.53iT - 31T^{2} \)
41 \( 1 + (-0.678 + 3.84i)T + (-38.5 - 14.0i)T^{2} \)
43 \( 1 - 7.61iT - 43T^{2} \)
47 \( 1 + (3.69 - 6.39i)T + (-23.5 - 40.7i)T^{2} \)
53 \( 1 + (2.83 + 1.03i)T + (40.6 + 34.0i)T^{2} \)
59 \( 1 + (3.91 - 10.7i)T + (-45.1 - 37.9i)T^{2} \)
61 \( 1 + (-3.06 - 0.541i)T + (57.3 + 20.8i)T^{2} \)
67 \( 1 + (-7.14 + 2.59i)T + (51.3 - 43.0i)T^{2} \)
71 \( 1 + (8.07 + 6.77i)T + (12.3 + 69.9i)T^{2} \)
73 \( 1 + 1.24T + 73T^{2} \)
79 \( 1 + (-1.62 - 4.45i)T + (-60.5 + 50.7i)T^{2} \)
83 \( 1 + (1.43 + 8.14i)T + (-77.9 + 28.3i)T^{2} \)
89 \( 1 + (-2.96 + 8.15i)T + (-68.1 - 57.2i)T^{2} \)
97 \( 1 + (-9.02 - 5.20i)T + (48.5 + 84.0i)T^{2} \)
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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

−11.44668881817780371698446627300, −10.60474746745037262159212781270, −9.981479903072857531178396721774, −8.991717676958352236251908522949, −7.66107299410699731651108010670, −6.68300839208996577585033815834, −6.15830318812431638587710424396, −5.18357314754989756535054595983, −3.04246229554220677459450866730, −2.72403129324044591491256505174, 0.07623912511788123088012000255, 2.16015886203572835714446259516, 3.94315894753178631497949565548, 4.90189356647386597544422828181, 5.69748877984763416786132093310, 6.94348547627505791444268744730, 7.79898111185971794761387603749, 9.369616284523603970803942298994, 9.739002736027531276018887506834, 10.22092056014824618752830091269

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