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Properties

Label 2-1413-157.156-c1-0-19
Degree $2$
Conductor $1413$
Sign $-0.274 - 0.961i$
Analytic cond. $11.2828$
Root an. cond. $3.35899$
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  + 2.55i·2-s − 4.50·4-s − 2.15i·5-s + 0.830i·7-s − 6.39i·8-s + 5.50·10-s − 5.63·11-s + 2.11·13-s − 2.11·14-s + 7.29·16-s + 5.31·17-s + 1.13·19-s + 9.72i·20-s − 14.3i·22-s + 0.00420i·23-s + ⋯
L(s)  = 1  + 1.80i·2-s − 2.25·4-s − 0.965i·5-s + 0.313i·7-s − 2.26i·8-s + 1.74·10-s − 1.69·11-s + 0.587·13-s − 0.566·14-s + 1.82·16-s + 1.28·17-s + 0.259·19-s + 2.17i·20-s − 3.06i·22-s + 0.000876i·23-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 1413 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1413 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.274 - 0.961i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(1413\)    =    \(3^{2} \cdot 157\)
Sign: $-0.274 - 0.961i$
Analytic conductor: \(11.2828\)
Root analytic conductor: \(3.35899\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{1413} (784, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 1413,\ (\ :1/2),\ -0.274 - 0.961i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.307542708\)
\(L(\frac12)\) \(\approx\) \(1.307542708\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad3 \( 1 \)
157 \( 1 + (3.44 + 12.0i)T \)
good2 \( 1 - 2.55iT - 2T^{2} \)
5 \( 1 + 2.15iT - 5T^{2} \)
7 \( 1 - 0.830iT - 7T^{2} \)
11 \( 1 + 5.63T + 11T^{2} \)
13 \( 1 - 2.11T + 13T^{2} \)
17 \( 1 - 5.31T + 17T^{2} \)
19 \( 1 - 1.13T + 19T^{2} \)
23 \( 1 - 0.00420iT - 23T^{2} \)
29 \( 1 + 7.71iT - 29T^{2} \)
31 \( 1 - 8.08T + 31T^{2} \)
37 \( 1 - 6.62T + 37T^{2} \)
41 \( 1 - 11.4iT - 41T^{2} \)
43 \( 1 - 8.86iT - 43T^{2} \)
47 \( 1 + 4.90T + 47T^{2} \)
53 \( 1 - 1.61iT - 53T^{2} \)
59 \( 1 - 0.544iT - 59T^{2} \)
61 \( 1 + 0.548iT - 61T^{2} \)
67 \( 1 + 7.34T + 67T^{2} \)
71 \( 1 - 14.3T + 71T^{2} \)
73 \( 1 + 1.25iT - 73T^{2} \)
79 \( 1 - 2.79iT - 79T^{2} \)
83 \( 1 + 7.18iT - 83T^{2} \)
89 \( 1 - 2.76T + 89T^{2} \)
97 \( 1 + 10.7iT - 97T^{2} \)
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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

−9.582633769479974678100209804680, −8.547359228053760185627220335789, −7.979665143465754435835345247238, −7.69377826644182102098579679118, −6.32095179472233900855019511426, −5.77594810082883408371702564826, −4.99445893559725275534449585129, −4.43056953104808068489795108369, −2.91960376680282842578670925133, −0.830836905625323781913685123164, 0.838623725899320497283342672873, 2.26197413154604148691621798428, 3.05814405910776656908752047638, 3.65551846584998367751003456833, 4.88411802284659862266769902706, 5.66334085817353599898993154737, 7.03604144377846012501181089586, 7.921929141174928098032061586866, 8.729693841556679984816147764631, 9.794747271239539935601309333519

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