Properties

Label 2-4001-1.1-c1-0-127
Degree $2$
Conductor $4001$
Sign $-1$
Analytic cond. $31.9481$
Root an. cond. $5.65226$
Motivic weight $1$
Arithmetic yes
Rational no
Primitive yes
Self-dual yes
Analytic rank $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 0.949·2-s − 2.99·3-s − 1.09·4-s − 0.667·5-s − 2.84·6-s − 3.07·7-s − 2.94·8-s + 5.96·9-s − 0.633·10-s + 0.288·11-s + 3.29·12-s − 5.99·13-s − 2.92·14-s + 1.99·15-s − 0.594·16-s + 2.27·17-s + 5.66·18-s + 6.30·19-s + 0.733·20-s + 9.21·21-s + 0.273·22-s + 6.81·23-s + 8.80·24-s − 4.55·25-s − 5.69·26-s − 8.87·27-s + 3.38·28-s + ⋯
L(s)  = 1  + 0.671·2-s − 1.72·3-s − 0.549·4-s − 0.298·5-s − 1.16·6-s − 1.16·7-s − 1.04·8-s + 1.98·9-s − 0.200·10-s + 0.0869·11-s + 0.949·12-s − 1.66·13-s − 0.780·14-s + 0.515·15-s − 0.148·16-s + 0.550·17-s + 1.33·18-s + 1.44·19-s + 0.163·20-s + 2.01·21-s + 0.0583·22-s + 1.42·23-s + 1.79·24-s − 0.910·25-s − 1.11·26-s − 1.70·27-s + 0.639·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4001 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4001\)
Sign: $-1$
Analytic conductor: \(31.9481\)
Root analytic conductor: \(5.65226\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad4001 \( 1+O(T) \)
good2 \( 1 - 0.949T + 2T^{2} \)
3 \( 1 + 2.99T + 3T^{2} \)
5 \( 1 + 0.667T + 5T^{2} \)
7 \( 1 + 3.07T + 7T^{2} \)
11 \( 1 - 0.288T + 11T^{2} \)
13 \( 1 + 5.99T + 13T^{2} \)
17 \( 1 - 2.27T + 17T^{2} \)
19 \( 1 - 6.30T + 19T^{2} \)
23 \( 1 - 6.81T + 23T^{2} \)
29 \( 1 - 3.07T + 29T^{2} \)
31 \( 1 + 5.87T + 31T^{2} \)
37 \( 1 - 8.77T + 37T^{2} \)
41 \( 1 + 2.99T + 41T^{2} \)
43 \( 1 - 3.17T + 43T^{2} \)
47 \( 1 - 2.56T + 47T^{2} \)
53 \( 1 - 10.2T + 53T^{2} \)
59 \( 1 + 12.9T + 59T^{2} \)
61 \( 1 + 11.7T + 61T^{2} \)
67 \( 1 - 7.70T + 67T^{2} \)
71 \( 1 + 3.96T + 71T^{2} \)
73 \( 1 - 10.9T + 73T^{2} \)
79 \( 1 - 2.13T + 79T^{2} \)
83 \( 1 + 6.40T + 83T^{2} \)
89 \( 1 - 1.99T + 89T^{2} \)
97 \( 1 - 17.5T + 97T^{2} \)
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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

−7.67473815366973575726283916870, −7.17919695805143424169643616085, −6.35740255188829622224223714203, −5.68514794737940019738583363804, −5.13614878452757173107460771847, −4.56549440987262855434385260346, −3.62122122326916420415535508258, −2.80162070950751674877516644194, −0.895843755082532398222088854785, 0, 0.895843755082532398222088854785, 2.80162070950751674877516644194, 3.62122122326916420415535508258, 4.56549440987262855434385260346, 5.13614878452757173107460771847, 5.68514794737940019738583363804, 6.35740255188829622224223714203, 7.17919695805143424169643616085, 7.67473815366973575726283916870

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