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Properties

Label 2-38e2-76.23-c0-0-1
Degree $2$
Conductor $1444$
Sign $0.963 - 0.267i$
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  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.473 − 0.397i)5-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.580 + 0.211i)10-s + (−0.280 + 1.59i)13-s + (0.173 + 0.984i)16-s + (1.52 + 0.553i)17-s + 18-s + 0.618·20-s + (−0.107 + 0.608i)25-s + (0.809 − 1.40i)26-s + (1.52 − 0.553i)29-s + (0.173 − 0.984i)32-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.766 + 0.642i)4-s + (0.473 − 0.397i)5-s + (−0.500 − 0.866i)8-s + (−0.939 + 0.342i)9-s + (−0.580 + 0.211i)10-s + (−0.280 + 1.59i)13-s + (0.173 + 0.984i)16-s + (1.52 + 0.553i)17-s + 18-s + 0.618·20-s + (−0.107 + 0.608i)25-s + (0.809 − 1.40i)26-s + (1.52 − 0.553i)29-s + (0.173 − 0.984i)32-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1444 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.963 - 0.267i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1444\)    =    \(2^{2} \cdot 19^{2}\)
Sign: $0.963 - 0.267i$
Analytic conductor: \(0.720649\)
Root analytic conductor: \(0.848910\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{1444} (99, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1444,\ (\ :0),\ 0.963 - 0.267i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(0.7382201982\)
\(L(\frac12)\) \(\approx\) \(0.7382201982\)
\(L(1)\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + (0.939 + 0.342i)T \)
19 \( 1 \)
good3 \( 1 + (0.939 - 0.342i)T^{2} \)
5 \( 1 + (-0.473 + 0.397i)T + (0.173 - 0.984i)T^{2} \)
7 \( 1 + (0.5 + 0.866i)T^{2} \)
11 \( 1 + (0.5 - 0.866i)T^{2} \)
13 \( 1 + (0.280 - 1.59i)T + (-0.939 - 0.342i)T^{2} \)
17 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
23 \( 1 + (-0.173 - 0.984i)T^{2} \)
29 \( 1 + (-1.52 + 0.553i)T + (0.766 - 0.642i)T^{2} \)
31 \( 1 + (0.5 + 0.866i)T^{2} \)
37 \( 1 - 0.618T + T^{2} \)
41 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
43 \( 1 + (-0.173 + 0.984i)T^{2} \)
47 \( 1 + (-0.766 + 0.642i)T^{2} \)
53 \( 1 + (-0.473 - 0.397i)T + (0.173 + 0.984i)T^{2} \)
59 \( 1 + (-0.766 - 0.642i)T^{2} \)
61 \( 1 + (1.23 + 1.04i)T + (0.173 + 0.984i)T^{2} \)
67 \( 1 + (-0.766 + 0.642i)T^{2} \)
71 \( 1 + (-0.173 + 0.984i)T^{2} \)
73 \( 1 + (-0.107 - 0.608i)T + (-0.939 + 0.342i)T^{2} \)
79 \( 1 + (0.939 - 0.342i)T^{2} \)
83 \( 1 + (0.5 + 0.866i)T^{2} \)
89 \( 1 + (-0.107 + 0.608i)T + (-0.939 - 0.342i)T^{2} \)
97 \( 1 + (-1.52 - 0.553i)T + (0.766 + 0.642i)T^{2} \)
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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

−9.682768973532610147347256220755, −9.039912127445472341223635810122, −8.293101623794104545501494889382, −7.61203695714711551966189002497, −6.54998974306443450148532894833, −5.83944118608165071152378528617, −4.70985171622714070091941623491, −3.47346839563428514799461255852, −2.40903223547916532589738432403, −1.37806062641709026246631063626, 0.912850641910967865228431253617, 2.62435094381009984568625753511, 3.15829891789116712813898812895, 5.07778629244403469487860978499, 5.78461283912455859771564388057, 6.39892520580659288600134741642, 7.49652950146497905819107035798, 8.058929506133913425033506855688, 8.849577931142348211055459580356, 9.753068678713352110389568489450

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