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Properties

Label 2-2432-8.5-c1-0-17
Degree $2$
Conductor $2432$
Sign $-i$
Analytic cond. $19.4196$
Root an. cond. $4.40676$
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  + 3.30i·3-s − 2.55i·5-s − 1.32·7-s − 7.91·9-s − 2.51i·11-s − 1.22i·13-s + 8.43·15-s + 0.210·17-s + i·19-s − 4.37i·21-s + 8.11·23-s − 1.51·25-s − 16.2i·27-s + 5.97i·29-s + 6.01·31-s + ⋯
L(s)  = 1  + 1.90i·3-s − 1.14i·5-s − 0.500·7-s − 2.63·9-s − 0.759i·11-s − 0.340i·13-s + 2.17·15-s + 0.0511·17-s + 0.229i·19-s − 0.955i·21-s + 1.69·23-s − 0.303·25-s − 3.12i·27-s + 1.10i·29-s + 1.08·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2432 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(2432\)    =    \(2^{7} \cdot 19\)
Sign: $-i$
Analytic conductor: \(19.4196\)
Root analytic conductor: \(4.40676\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{2432} (1217, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 2432,\ (\ :1/2),\ -i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.444535021\)
\(L(\frac12)\) \(\approx\) \(1.444535021\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
19 \( 1 - iT \)
good3 \( 1 - 3.30iT - 3T^{2} \)
5 \( 1 + 2.55iT - 5T^{2} \)
7 \( 1 + 1.32T + 7T^{2} \)
11 \( 1 + 2.51iT - 11T^{2} \)
13 \( 1 + 1.22iT - 13T^{2} \)
17 \( 1 - 0.210T + 17T^{2} \)
23 \( 1 - 8.11T + 23T^{2} \)
29 \( 1 - 5.97iT - 29T^{2} \)
31 \( 1 - 6.01T + 31T^{2} \)
37 \( 1 - 11.2iT - 37T^{2} \)
41 \( 1 + 0.996T + 41T^{2} \)
43 \( 1 - 7.83iT - 43T^{2} \)
47 \( 1 + 0.910T + 47T^{2} \)
53 \( 1 - 3.32iT - 53T^{2} \)
59 \( 1 - 2.04iT - 59T^{2} \)
61 \( 1 + 10.0iT - 61T^{2} \)
67 \( 1 - 11.7iT - 67T^{2} \)
71 \( 1 - 11.9T + 71T^{2} \)
73 \( 1 + 13.0T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 + 4.50iT - 83T^{2} \)
89 \( 1 - 6.64T + 89T^{2} \)
97 \( 1 - 9.92T + 97T^{2} \)
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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

−9.154567481372341002318994782491, −8.643572705128067882962357341003, −8.085105085334344355476758578617, −6.56537916406091036465291654515, −5.72046996973131223864981125098, −4.86164250125877924754932060799, −4.66876659128880328125338747123, −3.36737217057681495124019824404, −3.02907981440682553021919903631, −0.917711429200354268084260935795, 0.63639915598929545403730598785, 2.01898586677790991040834096812, 2.63105303841434102530126684793, 3.49960200887249472152433107857, 5.00625816624093494286547087281, 6.12049001277100212229067941690, 6.58114874971994313994426730297, 7.22712920611391412645839379554, 7.57020070604244800401130578664, 8.609945645304460064729860845311

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