Properties

Label 2-4004-13.12-c1-0-16
Degree $2$
Conductor $4004$
Sign $0.677 - 0.735i$
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  + 0.334·3-s − 1.98i·5-s + i·7-s − 2.88·9-s i·11-s + (−2.44 + 2.65i)13-s − 0.664i·15-s − 5.84·17-s − 2.86i·19-s + 0.334i·21-s + 0.564·23-s + 1.04·25-s − 1.96·27-s + 8.26·29-s + 1.77i·31-s + ⋯
L(s)  = 1  + 0.192·3-s − 0.889i·5-s + 0.377i·7-s − 0.962·9-s − 0.301i·11-s + (−0.677 + 0.735i)13-s − 0.171i·15-s − 1.41·17-s − 0.658i·19-s + 0.0728i·21-s + 0.117·23-s + 0.208·25-s − 0.378·27-s + 1.53·29-s + 0.317i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.677 - 0.735i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $0.677 - 0.735i$
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: \((2,\ 4004,\ (\ :1/2),\ 0.677 - 0.735i)\)

Particular Values

\(L(1)\) \(\approx\) \(1.258480446\)
\(L(\frac12)\) \(\approx\) \(1.258480446\)
\(L(\frac{3}{2})\) 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 \( 1 + (2.44 - 2.65i)T \)
good3 \( 1 - 0.334T + 3T^{2} \)
5 \( 1 + 1.98iT - 5T^{2} \)
17 \( 1 + 5.84T + 17T^{2} \)
19 \( 1 + 2.86iT - 19T^{2} \)
23 \( 1 - 0.564T + 23T^{2} \)
29 \( 1 - 8.26T + 29T^{2} \)
31 \( 1 - 1.77iT - 31T^{2} \)
37 \( 1 + 0.265iT - 37T^{2} \)
41 \( 1 - 5.63iT - 41T^{2} \)
43 \( 1 + 0.932T + 43T^{2} \)
47 \( 1 - 10.4iT - 47T^{2} \)
53 \( 1 - 11.6T + 53T^{2} \)
59 \( 1 + 8.63iT - 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 + 1.98iT - 67T^{2} \)
71 \( 1 - 10.5iT - 71T^{2} \)
73 \( 1 - 3.74iT - 73T^{2} \)
79 \( 1 - 4.24T + 79T^{2} \)
83 \( 1 - 8.15iT - 83T^{2} \)
89 \( 1 - 18.0iT - 89T^{2} \)
97 \( 1 - 12.1iT - 97T^{2} \)
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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

−8.492635255366390395037343553644, −8.197572849358985555778143010711, −6.90908207377607833969319008421, −6.46019317579518847489332743987, −5.39779663074453622410005247775, −4.83739672324836004814496736411, −4.12981466477653580758556856580, −2.84497253617170422166334295201, −2.28752108907896395610581335456, −0.903828986076521221328624855859, 0.42179918456329904758092609867, 2.14735599429111218578474462550, 2.77063560131021997204006486134, 3.59313032483552382695288950686, 4.54641649803523170224788040852, 5.40695808180982662479538770748, 6.24594276008378044014029258862, 6.95938065725382470026267844968, 7.49899560745778720760916694669, 8.465979132742207240604973580636

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