Properties

Label 2-4004-77.76-c1-0-63
Degree $2$
Conductor $4004$
Sign $-0.528 + 0.849i$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
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  − 3.23i·3-s + 0.717i·5-s + (0.284 − 2.63i)7-s − 7.45·9-s + (2.98 + 1.43i)11-s − 13-s + 2.31·15-s + 2.69·17-s + 5.82·19-s + (−8.50 − 0.919i)21-s + 8.92·23-s + 4.48·25-s + 14.3i·27-s − 1.83i·29-s − 0.614i·31-s + ⋯
L(s)  = 1  − 1.86i·3-s + 0.320i·5-s + (0.107 − 0.994i)7-s − 2.48·9-s + (0.900 + 0.433i)11-s − 0.277·13-s + 0.598·15-s + 0.653·17-s + 1.33·19-s + (−1.85 − 0.200i)21-s + 1.86·23-s + 0.897·25-s + 2.77i·27-s − 0.340i·29-s − 0.110i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.528 + 0.849i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.528 + 0.849i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (3849, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ -0.528 + 0.849i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.241814737\)
\(L(\frac12)\) \(\approx\) \(2.241814737\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 + (-0.284 + 2.63i)T \)
11 \( 1 + (-2.98 - 1.43i)T \)
13 \( 1 + T \)
good3 \( 1 + 3.23iT - 3T^{2} \)
5 \( 1 - 0.717iT - 5T^{2} \)
17 \( 1 - 2.69T + 17T^{2} \)
19 \( 1 - 5.82T + 19T^{2} \)
23 \( 1 - 8.92T + 23T^{2} \)
29 \( 1 + 1.83iT - 29T^{2} \)
31 \( 1 + 0.614iT - 31T^{2} \)
37 \( 1 - 10.1T + 37T^{2} \)
41 \( 1 - 9.05T + 41T^{2} \)
43 \( 1 + 7.96iT - 43T^{2} \)
47 \( 1 - 3.16iT - 47T^{2} \)
53 \( 1 - 6.50T + 53T^{2} \)
59 \( 1 - 4.87iT - 59T^{2} \)
61 \( 1 + 11.0T + 61T^{2} \)
67 \( 1 + 0.420T + 67T^{2} \)
71 \( 1 + 3.11T + 71T^{2} \)
73 \( 1 + 10.9T + 73T^{2} \)
79 \( 1 - 14.7iT - 79T^{2} \)
83 \( 1 + 9.16T + 83T^{2} \)
89 \( 1 + 15.4iT - 89T^{2} \)
97 \( 1 - 3.21iT - 97T^{2} \)
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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

−7.78437523374961367083973803814, −7.31479456529383265677031333367, −7.04587615205391768024883416738, −6.23895041885364153104460339799, −5.45788611228414050032788904160, −4.41088545010361396329423351373, −3.23027966215639861942018251097, −2.56859527305195416203589705897, −1.23265134449985263565071127005, −0.921781722869329883034932842734, 1.06876960011193142507997815442, 2.95729763558471910023336252121, 3.09448673392311764686738209592, 4.31307990267370927262769882052, 4.87622821039712904105837146710, 5.54395699077592681838997584321, 6.09814661703580747551496452548, 7.31567005656005704201365077977, 8.350743302088698910689217733693, 8.950899837670401253962103564753

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