Properties

Label 2-4004-77.76-c1-0-84
Degree $2$
Conductor $4004$
Sign $-0.786 + 0.617i$
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  + 0.151i·3-s − 3.45i·5-s + (2.64 + 0.100i)7-s + 2.97·9-s + (−2.52 + 2.14i)11-s + 13-s + 0.524·15-s − 4.73·17-s + 1.06·19-s + (−0.0151 + 0.401i)21-s − 8.86·23-s − 6.93·25-s + 0.907i·27-s − 0.523i·29-s − 7.62i·31-s + ⋯
L(s)  = 1  + 0.0876i·3-s − 1.54i·5-s + (0.999 + 0.0378i)7-s + 0.992·9-s + (−0.762 + 0.646i)11-s + 0.277·13-s + 0.135·15-s − 1.14·17-s + 0.243·19-s + (−0.00331 + 0.0875i)21-s − 1.84·23-s − 1.38·25-s + 0.174i·27-s − 0.0971i·29-s − 1.36i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.786 + 0.617i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.786 + 0.617i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.786 + 0.617i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.331493631\)
\(L(\frac12)\) \(\approx\) \(1.331493631\)
\(L(\frac{3}{2})\) 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 \( 1 + (-2.64 - 0.100i)T \)
11 \( 1 + (2.52 - 2.14i)T \)
13 \( 1 - T \)
good3 \( 1 - 0.151iT - 3T^{2} \)
5 \( 1 + 3.45iT - 5T^{2} \)
17 \( 1 + 4.73T + 17T^{2} \)
19 \( 1 - 1.06T + 19T^{2} \)
23 \( 1 + 8.86T + 23T^{2} \)
29 \( 1 + 0.523iT - 29T^{2} \)
31 \( 1 + 7.62iT - 31T^{2} \)
37 \( 1 + 5.23T + 37T^{2} \)
41 \( 1 + 0.443T + 41T^{2} \)
43 \( 1 + 10.5iT - 43T^{2} \)
47 \( 1 + 6.76iT - 47T^{2} \)
53 \( 1 + 1.69T + 53T^{2} \)
59 \( 1 + 3.84iT - 59T^{2} \)
61 \( 1 + 6.13T + 61T^{2} \)
67 \( 1 + 10.8T + 67T^{2} \)
71 \( 1 - 6.85T + 71T^{2} \)
73 \( 1 - 8.18T + 73T^{2} \)
79 \( 1 + 5.60iT - 79T^{2} \)
83 \( 1 + 2.69T + 83T^{2} \)
89 \( 1 - 11.9iT - 89T^{2} \)
97 \( 1 - 6.19iT - 97T^{2} \)
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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

−8.111245007748532032061966464015, −7.70529016055064368379121586380, −6.77683050246456368540058924662, −5.65337849403713854640831827270, −5.08097324841754524516986170765, −4.35290733956582721662520967568, −3.98406679963271690651339330116, −2.10586019938130780217598936497, −1.70845433696248247951720753885, −0.35630963680520834999928491199, 1.50473689233386800366341858900, 2.37052851050805731922872598091, 3.25006519026894451346382450235, 4.15075117349164184527311790663, 4.92143128346811775109619625555, 5.98527791566668836039240175722, 6.56099449861890760512613333326, 7.33018741138001612355725642450, 7.87047940895136239407948278013, 8.533413529949819631127832735126

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