Properties

Label 2-4004-13.12-c1-0-39
Degree $2$
Conductor $4004$
Sign $0.982 - 0.187i$
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  + 0.705·3-s + 2.65i·5-s i·7-s − 2.50·9-s i·11-s + (3.54 − 0.674i)13-s + 1.87i·15-s − 0.198·17-s − 5.34i·19-s − 0.705i·21-s + 6.13·23-s − 2.05·25-s − 3.88·27-s + 2.90·29-s + 5.55i·31-s + ⋯
L(s)  = 1  + 0.407·3-s + 1.18i·5-s − 0.377i·7-s − 0.834·9-s − 0.301i·11-s + (0.982 − 0.187i)13-s + 0.483i·15-s − 0.0481·17-s − 1.22i·19-s − 0.153i·21-s + 1.28·23-s − 0.410·25-s − 0.747·27-s + 0.539·29-s + 0.998i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 - 0.187i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.982 - 0.187i$
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: \((2,\ 4004,\ (\ :1/2),\ 0.982 - 0.187i)\)

Particular Values

\(L(1)\) \(\approx\) \(2.144939967\)
\(L(\frac12)\) \(\approx\) \(2.144939967\)
\(L(\frac{3}{2})\) 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 \( 1 + (-3.54 + 0.674i)T \)
good3 \( 1 - 0.705T + 3T^{2} \)
5 \( 1 - 2.65iT - 5T^{2} \)
17 \( 1 + 0.198T + 17T^{2} \)
19 \( 1 + 5.34iT - 19T^{2} \)
23 \( 1 - 6.13T + 23T^{2} \)
29 \( 1 - 2.90T + 29T^{2} \)
31 \( 1 - 5.55iT - 31T^{2} \)
37 \( 1 + 3.81iT - 37T^{2} \)
41 \( 1 - 0.304iT - 41T^{2} \)
43 \( 1 + 2.09T + 43T^{2} \)
47 \( 1 - 3.81iT - 47T^{2} \)
53 \( 1 - 4.43T + 53T^{2} \)
59 \( 1 + 14.5iT - 59T^{2} \)
61 \( 1 - 9.96T + 61T^{2} \)
67 \( 1 - 6.86iT - 67T^{2} \)
71 \( 1 + 14.7iT - 71T^{2} \)
73 \( 1 - 11.4iT - 73T^{2} \)
79 \( 1 + 4.78T + 79T^{2} \)
83 \( 1 + 15.0iT - 83T^{2} \)
89 \( 1 - 7.28iT - 89T^{2} \)
97 \( 1 - 13.8iT - 97T^{2} \)
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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

−8.617738838754465159639526810002, −7.70290406558088931021351461285, −6.91029398990809491430363997328, −6.48299267439748212626426984275, −5.57292735669921815476730565821, −4.69379464222490335361698285841, −3.44473340059270393317476386855, −3.15292787382139147219953961768, −2.27375458026307832020649236684, −0.790078012668799334727735031853, 0.880266726685103744072712724775, 1.87926183071185865139746704436, 2.94330937066251421445561800518, 3.83193535139042631244196272302, 4.66301445781203648768967393778, 5.51346613524663474011194347872, 5.99916676773707291287880116291, 7.01074029333536720642855135128, 8.007603644011798100270592039708, 8.571041809961875323270430181824

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