Properties

Label 2-4004-13.12-c1-0-34
Degree $2$
Conductor $4004$
Sign $0.0495 + 0.998i$
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  − 2.50·3-s + 2.36i·5-s + i·7-s + 3.25·9-s + i·11-s + (0.178 + 3.60i)13-s − 5.92i·15-s − 2.97·17-s − 0.164i·19-s − 2.50i·21-s − 8.05·23-s − 0.613·25-s − 0.649·27-s − 4.68·29-s − 1.30i·31-s + ⋯
L(s)  = 1  − 1.44·3-s + 1.05i·5-s + 0.377i·7-s + 1.08·9-s + 0.301i·11-s + (0.0495 + 0.998i)13-s − 1.53i·15-s − 0.720·17-s − 0.0377i·19-s − 0.545i·21-s − 1.67·23-s − 0.122·25-s − 0.124·27-s − 0.870·29-s − 0.234i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0495 + 0.998i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0495 + 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.0495 + 0.998i$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: $\chi_{4004} (2157, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 0.0495 + 0.998i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.02606514161\)
\(L(\frac12)\) \(\approx\) \(0.02606514161\)
\(L(\frac{3}{2})\) 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 \( 1 - iT \)
11 \( 1 - iT \)
13 \( 1 + (-0.178 - 3.60i)T \)
good3 \( 1 + 2.50T + 3T^{2} \)
5 \( 1 - 2.36iT - 5T^{2} \)
17 \( 1 + 2.97T + 17T^{2} \)
19 \( 1 + 0.164iT - 19T^{2} \)
23 \( 1 + 8.05T + 23T^{2} \)
29 \( 1 + 4.68T + 29T^{2} \)
31 \( 1 + 1.30iT - 31T^{2} \)
37 \( 1 + 1.83iT - 37T^{2} \)
41 \( 1 - 4.62iT - 41T^{2} \)
43 \( 1 + 7.77T + 43T^{2} \)
47 \( 1 + 7.81iT - 47T^{2} \)
53 \( 1 - 12.6T + 53T^{2} \)
59 \( 1 - 7.01iT - 59T^{2} \)
61 \( 1 + 8.99T + 61T^{2} \)
67 \( 1 + 3.03iT - 67T^{2} \)
71 \( 1 - 13.5iT - 71T^{2} \)
73 \( 1 - 12.8iT - 73T^{2} \)
79 \( 1 + 10.0T + 79T^{2} \)
83 \( 1 + 11.4iT - 83T^{2} \)
89 \( 1 - 18.0iT - 89T^{2} \)
97 \( 1 + 4.72iT - 97T^{2} \)
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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

−8.212247477898736492965813603309, −7.07614563890967158127037503714, −6.83190197661481751138623164926, −6.03389842848003124522500254200, −5.52777970522861691654907709379, −4.51400676972737683128264000120, −3.86168282894036078186272026013, −2.58099446108863873203704355634, −1.71183051197886866932108124884, −0.01258819015431853501179374113, 0.78131231653728150804258235457, 1.87136090724412353735112407119, 3.38563471060042588908031367331, 4.38702089167551371307146251618, 4.93398681337310507908010136657, 5.72170743509252405940447650486, 6.12782058589972600005598397527, 7.06028232424605206680547734547, 7.912203778295018739353408538562, 8.561239639205019121915356830992

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