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Properties

Label 2-38e2-76.47-c0-0-2
Degree $2$
Conductor $1444$
Sign $-0.588 + 0.808i$
Analytic cond. $0.720649$
Root an. cond. $0.848910$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.580 − 0.211i)5-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.107 + 0.608i)10-s + (1.23 + 1.04i)13-s + (0.766 − 0.642i)16-s + (−0.280 − 1.59i)17-s − 0.999·18-s + 0.618·20-s + (−0.473 − 0.397i)25-s + (0.809 − 1.40i)26-s + (0.280 − 1.59i)29-s + (−0.766 − 0.642i)32-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (−0.580 − 0.211i)5-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (−0.107 + 0.608i)10-s + (1.23 + 1.04i)13-s + (0.766 − 0.642i)16-s + (−0.280 − 1.59i)17-s − 0.999·18-s + 0.618·20-s + (−0.473 − 0.397i)25-s + (0.809 − 1.40i)26-s + (0.280 − 1.59i)29-s + (−0.766 − 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.588 + 0.808i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $-0.588 + 0.808i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (1111, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ -0.588 + 0.808i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7892921913\)
\(L(\frac12)\) \(\approx\) \(0.7892921913\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.173 + 0.984i)T \)
19 \( 1 \)
good3 \( 1 + (-0.173 + 0.984i)T^{2} \)
5 \( 1 + (0.580 + 0.211i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (-1.23 - 1.04i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (0.280 + 1.59i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (-0.280 + 1.59i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 + 0.618T + T^{2} \)
41 \( 1 + (0.473 - 0.397i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (-0.580 + 0.211i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (0.473 + 0.397i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (-0.280 - 1.59i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.495654843052841644384997393804, −8.803979149957232278030275185521, −8.150090404092446378062977004859, −7.07902520882376314976997673312, −6.19625705618429246702690840777, −4.91787664134058225518641985313, −4.06822445332037142171692583192, −3.45892713906433069862922312026, −2.17575970016404424170055900549, −0.76504225728351828022175193910, 1.52120590255491963475897268487, 3.37837723680687878522246748893, 4.13750298036742834755797425335, 5.24295351937689121821143267917, 5.90783422645155030838302098086, 6.86313568511966755388987856217, 7.64785213243947434071355428872, 8.377791185579993264762821142071, 8.724992647485513286434188361991, 10.08068085650480802032934334325

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