Properties

Label 2-4004-13.12-c1-0-49
Degree $2$
Conductor $4004$
Sign $-0.331 + 0.943i$
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  − 0.0797·3-s + 0.579i·5-s i·7-s − 2.99·9-s i·11-s + (−1.19 + 3.40i)13-s − 0.0461i·15-s + 6.63·17-s − 3.33i·19-s + 0.0797i·21-s − 8.63·23-s + 4.66·25-s + 0.477·27-s + 0.664·29-s + 0.646i·31-s + ⋯
L(s)  = 1  − 0.0460·3-s + 0.258i·5-s − 0.377i·7-s − 0.997·9-s − 0.301i·11-s + (−0.331 + 0.943i)13-s − 0.0119i·15-s + 1.61·17-s − 0.764i·19-s + 0.0173i·21-s − 1.79·23-s + 0.932·25-s + 0.0919·27-s + 0.123·29-s + 0.116i·31-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.331 + 0.943i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $-0.331 + 0.943i$
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: \((2,\ 4004,\ (\ :1/2),\ -0.331 + 0.943i)\)

Particular Values

\(L(1)\) \(\approx\) \(0.8735706217\)
\(L(\frac12)\) \(\approx\) \(0.8735706217\)
\(L(\frac{3}{2})\) 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 \( 1 + (1.19 - 3.40i)T \)
good3 \( 1 + 0.0797T + 3T^{2} \)
5 \( 1 - 0.579iT - 5T^{2} \)
17 \( 1 - 6.63T + 17T^{2} \)
19 \( 1 + 3.33iT - 19T^{2} \)
23 \( 1 + 8.63T + 23T^{2} \)
29 \( 1 - 0.664T + 29T^{2} \)
31 \( 1 - 0.646iT - 31T^{2} \)
37 \( 1 - 0.0692iT - 37T^{2} \)
41 \( 1 + 0.306iT - 41T^{2} \)
43 \( 1 + 5.52T + 43T^{2} \)
47 \( 1 - 2.26iT - 47T^{2} \)
53 \( 1 - 4.29T + 53T^{2} \)
59 \( 1 + 11.0iT - 59T^{2} \)
61 \( 1 + 2.98T + 61T^{2} \)
67 \( 1 + 2.33iT - 67T^{2} \)
71 \( 1 - 3.49iT - 71T^{2} \)
73 \( 1 + 9.99iT - 73T^{2} \)
79 \( 1 + 12.0T + 79T^{2} \)
83 \( 1 + 9.43iT - 83T^{2} \)
89 \( 1 + 6.81iT - 89T^{2} \)
97 \( 1 + 14.5iT - 97T^{2} \)
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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

−8.206705470873189196656806927906, −7.53739018958279996658923428232, −6.72143889226714179099435578993, −6.04185839605392275898966065552, −5.29186728184344422530876301297, −4.43707531234459214659885487785, −3.47390759336104023512928489425, −2.78874424467150925592293133883, −1.65801452088506252664187787129, −0.26857893961171878438114092613, 1.13257263793074191872450730827, 2.38675806651399168795622734862, 3.17853360312447480362207780842, 4.02716246833900386273452973869, 5.22444710942714779927700398450, 5.58584971262915994876599475284, 6.27154221917607384167949063426, 7.37224276723410274226363933487, 8.117920005884189358472644048400, 8.419105322003407006376174898173

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