Properties

Label 2-4006-1.1-c1-0-50
Degree $2$
Conductor $4006$
Sign $1$
Analytic cond. $31.9880$
Root an. cond. $5.65579$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.676·3-s + 4-s − 1.65·5-s − 0.676·6-s + 3.29·7-s + 8-s − 2.54·9-s − 1.65·10-s + 5.58·11-s − 0.676·12-s − 0.428·13-s + 3.29·14-s + 1.11·15-s + 16-s + 3.61·17-s − 2.54·18-s + 3.36·19-s − 1.65·20-s − 2.23·21-s + 5.58·22-s + 0.496·23-s − 0.676·24-s − 2.26·25-s − 0.428·26-s + 3.74·27-s + 3.29·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.390·3-s + 0.5·4-s − 0.739·5-s − 0.276·6-s + 1.24·7-s + 0.353·8-s − 0.847·9-s − 0.523·10-s + 1.68·11-s − 0.195·12-s − 0.118·13-s + 0.881·14-s + 0.289·15-s + 0.250·16-s + 0.875·17-s − 0.599·18-s + 0.772·19-s − 0.369·20-s − 0.487·21-s + 1.19·22-s + 0.103·23-s − 0.138·24-s − 0.452·25-s − 0.0839·26-s + 0.721·27-s + 0.623·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4006\)    =    \(2 \cdot 2003\)
Sign: $1$
Analytic conductor: \(31.9880\)
Root analytic conductor: \(5.65579\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4006,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(2.879030163\)
\(L(\frac12)\) \(\approx\) \(2.879030163\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
2003 \( 1 + T \)
good3 \( 1 + 0.676T + 3T^{2} \)
5 \( 1 + 1.65T + 5T^{2} \)
7 \( 1 - 3.29T + 7T^{2} \)
11 \( 1 - 5.58T + 11T^{2} \)
13 \( 1 + 0.428T + 13T^{2} \)
17 \( 1 - 3.61T + 17T^{2} \)
19 \( 1 - 3.36T + 19T^{2} \)
23 \( 1 - 0.496T + 23T^{2} \)
29 \( 1 + 2.59T + 29T^{2} \)
31 \( 1 + 8.75T + 31T^{2} \)
37 \( 1 - 1.22T + 37T^{2} \)
41 \( 1 - 3.45T + 41T^{2} \)
43 \( 1 - 9.11T + 43T^{2} \)
47 \( 1 + 2.38T + 47T^{2} \)
53 \( 1 - 3.55T + 53T^{2} \)
59 \( 1 - 6.55T + 59T^{2} \)
61 \( 1 - 10.1T + 61T^{2} \)
67 \( 1 - 0.840T + 67T^{2} \)
71 \( 1 + 15.5T + 71T^{2} \)
73 \( 1 - 2.78T + 73T^{2} \)
79 \( 1 + 12.8T + 79T^{2} \)
83 \( 1 + 9.96T + 83T^{2} \)
89 \( 1 - 1.84T + 89T^{2} \)
97 \( 1 - 11.8T + 97T^{2} \)
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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

−8.368043647630027451459297757754, −7.50244800001510126403051003592, −7.11790503502283514722106245464, −5.88186888862666576581883786636, −5.58585909590680858442705460826, −4.61415995525263253415827168370, −3.93819129795867708132344821521, −3.25005751365221003888333620971, −1.94904456621015245371700198214, −0.943678577942730376577385714851, 0.943678577942730376577385714851, 1.94904456621015245371700198214, 3.25005751365221003888333620971, 3.93819129795867708132344821521, 4.61415995525263253415827168370, 5.58585909590680858442705460826, 5.88186888862666576581883786636, 7.11790503502283514722106245464, 7.50244800001510126403051003592, 8.368043647630027451459297757754

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