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Properties

Label 2-38e2-76.47-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.486 + 0.873i$
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 + (1.52 + 0.553i)5-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.280 − 1.59i)10-s + (−0.473 − 0.397i)13-s + (0.766 − 0.642i)16-s + (0.107 + 0.608i)17-s − 0.999·18-s − 1.61·20-s + (1.23 + 1.04i)25-s + (−0.309 + 0.535i)26-s + (−0.107 + 0.608i)29-s + (−0.766 − 0.642i)32-s + ⋯
L(s)  = 1  + (−0.173 − 0.984i)2-s + (−0.939 + 0.342i)4-s + (1.52 + 0.553i)5-s + (0.5 + 0.866i)8-s + (0.173 − 0.984i)9-s + (0.280 − 1.59i)10-s + (−0.473 − 0.397i)13-s + (0.766 − 0.642i)16-s + (0.107 + 0.608i)17-s − 0.999·18-s − 1.61·20-s + (1.23 + 1.04i)25-s + (−0.309 + 0.535i)26-s + (−0.107 + 0.608i)29-s + (−0.766 − 0.642i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.486 + 0.873i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.486 + 0.873i$
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: \((2,\ 1444,\ (\ :0),\ 0.486 + 0.873i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(1.191119298\)
\(L(\frac12)\) \(\approx\) \(1.191119298\)
\(L(1)\) 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 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.473 + 0.397i)T + (0.173 + 0.984i)T^{2} \)
17 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
23 \( 1 + (-0.766 + 0.642i)T^{2} \)
29 \( 1 + (0.107 - 0.608i)T + (-0.939 - 0.342i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 1.61T + T^{2} \)
41 \( 1 + (-1.23 + 1.04i)T + (0.173 - 0.984i)T^{2} \)
43 \( 1 + (-0.766 - 0.642i)T^{2} \)
47 \( 1 + (0.939 + 0.342i)T^{2} \)
53 \( 1 + (1.52 - 0.553i)T + (0.766 - 0.642i)T^{2} \)
59 \( 1 + (0.939 - 0.342i)T^{2} \)
61 \( 1 + (0.580 - 0.211i)T + (0.766 - 0.642i)T^{2} \)
67 \( 1 + (0.939 + 0.342i)T^{2} \)
71 \( 1 + (-0.766 - 0.642i)T^{2} \)
73 \( 1 + (1.23 - 1.04i)T + (0.173 - 0.984i)T^{2} \)
79 \( 1 + (-0.173 + 0.984i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-1.23 - 1.04i)T + (0.173 + 0.984i)T^{2} \)
97 \( 1 + (0.107 + 0.608i)T + (-0.939 + 0.342i)T^{2} \)
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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

−9.494338238435604446398431468483, −9.353502033758122239818988117750, −8.213127464883290743566880873220, −7.14368547156936549599215152178, −6.14442237819495330014886018958, −5.49987100529981203361772358070, −4.32896981252041118646426137327, −3.22548490183505783395225855370, −2.41574862324661221882492003024, −1.33185650954271832206818610862, 1.44165963038130625532282986764, 2.60288029160539535657622058719, 4.47674225529545727972770462152, 4.95649827089980103883908412616, 5.85817761014699889423004075525, 6.42565472483650535477958846262, 7.53960310364801222676985530762, 8.107607064305884285592160650511, 9.359436295449473931262726769049, 9.429808463221714953803934245684

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