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Properties

Label 2-444-444.443-c1-0-42
Degree $2$
Conductor $444$
Sign $0.991 - 0.130i$
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 + 1.32i)2-s + 1.73·3-s + (−1.50 − 1.32i)4-s − 2·5-s + (−0.866 + 2.29i)6-s − 2.64i·7-s + (2.50 − 1.32i)8-s + 2.99·9-s + (1 − 2.64i)10-s + 1.73·11-s + (−2.59 − 2.29i)12-s − 4.58i·13-s + (3.50 + 1.32i)14-s − 3.46·15-s + (0.500 + 3.96i)16-s + 5·17-s + ⋯
L(s)  = 1  + (−0.353 + 0.935i)2-s + 1.00·3-s + (−0.750 − 0.661i)4-s − 0.894·5-s + (−0.353 + 0.935i)6-s − 0.999i·7-s + (0.883 − 0.467i)8-s + 0.999·9-s + (0.316 − 0.836i)10-s + 0.522·11-s + (−0.749 − 0.661i)12-s − 1.27i·13-s + (0.935 + 0.353i)14-s − 0.894·15-s + (0.125 + 0.992i)16-s + 1.21·17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 444 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.991 - 0.130i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(444\)    =    \(2^{2} \cdot 3 \cdot 37\)
Sign: $0.991 - 0.130i$
Analytic conductor: \(3.54535\)
Root analytic conductor: \(1.88291\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{444} (443, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 444,\ (\ :1/2),\ 0.991 - 0.130i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.37175 + 0.0895921i\)
\(L(\frac12)\) \(\approx\) \(1.37175 + 0.0895921i\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.5 - 1.32i)T \)
3 \( 1 - 1.73T \)
37 \( 1 + (-4 + 4.58i)T \)
good5 \( 1 + 2T + 5T^{2} \)
7 \( 1 + 2.64iT - 7T^{2} \)
11 \( 1 - 1.73T + 11T^{2} \)
13 \( 1 + 4.58iT - 13T^{2} \)
17 \( 1 - 5T + 17T^{2} \)
19 \( 1 - 1.73T + 19T^{2} \)
23 \( 1 - 2.64iT - 23T^{2} \)
29 \( 1 - 4T + 29T^{2} \)
31 \( 1 + 3.46T + 31T^{2} \)
41 \( 1 + 9.16iT - 41T^{2} \)
43 \( 1 + 3.46T + 43T^{2} \)
47 \( 1 + 10.3T + 47T^{2} \)
53 \( 1 - 13.7iT - 53T^{2} \)
59 \( 1 - 5.29iT - 59T^{2} \)
61 \( 1 + 9.16iT - 61T^{2} \)
67 \( 1 - 10.5iT - 67T^{2} \)
71 \( 1 + 3.46T + 71T^{2} \)
73 \( 1 - 9T + 73T^{2} \)
79 \( 1 - 13.8T + 79T^{2} \)
83 \( 1 + 8.66T + 83T^{2} \)
89 \( 1 + 5T + 89T^{2} \)
97 \( 1 - 97T^{2} \)
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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

−10.74696595935847494321063150306, −10.02253127526084546712467865045, −9.173071474102844922303392784635, −7.976470378098627828228136103341, −7.74669951575557378723892157516, −6.91233841949818088864544294423, −5.46500310483274352148260363931, −4.14526590795806474826433952722, −3.41933303187025742582987850779, −1.02466463393458552628768295002, 1.64037809551906628826303055811, 2.94629111260902061534028494665, 3.83955061595626595731518339034, 4.85559875788224515695139946442, 6.71402471687749042014481399163, 7.943358202225609909430796999182, 8.418420262981741253942001695208, 9.396292262218265264182515423860, 9.888542693547246233829576286172, 11.33113044217958357347309549913

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