Properties

Label 2-4001-1.1-c1-0-128
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 $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 0.892·2-s − 0.475·3-s − 1.20·4-s + 4.01·5-s + 0.424·6-s − 0.567·7-s + 2.85·8-s − 2.77·9-s − 3.58·10-s + 4.71·11-s + 0.571·12-s + 0.480·13-s + 0.506·14-s − 1.90·15-s − 0.146·16-s + 3.16·17-s + 2.47·18-s + 4.33·19-s − 4.82·20-s + 0.269·21-s − 4.20·22-s − 6.76·23-s − 1.35·24-s + 11.1·25-s − 0.429·26-s + 2.74·27-s + 0.683·28-s + ⋯
L(s)  = 1  − 0.631·2-s − 0.274·3-s − 0.601·4-s + 1.79·5-s + 0.173·6-s − 0.214·7-s + 1.01·8-s − 0.924·9-s − 1.13·10-s + 1.42·11-s + 0.164·12-s + 0.133·13-s + 0.135·14-s − 0.492·15-s − 0.0366·16-s + 0.767·17-s + 0.583·18-s + 0.994·19-s − 1.07·20-s + 0.0588·21-s − 0.897·22-s − 1.41·23-s − 0.277·24-s + 2.22·25-s − 0.0841·26-s + 0.527·27-s + 0.129·28-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: \(0\)
Selberg data: \((2,\ 4001,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.628517311\)
\(L(\frac12)\) \(\approx\) \(1.628517311\)
\(L(\frac{3}{2})\) 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 \( 1 + 0.892T + 2T^{2} \)
3 \( 1 + 0.475T + 3T^{2} \)
5 \( 1 - 4.01T + 5T^{2} \)
7 \( 1 + 0.567T + 7T^{2} \)
11 \( 1 - 4.71T + 11T^{2} \)
13 \( 1 - 0.480T + 13T^{2} \)
17 \( 1 - 3.16T + 17T^{2} \)
19 \( 1 - 4.33T + 19T^{2} \)
23 \( 1 + 6.76T + 23T^{2} \)
29 \( 1 - 8.94T + 29T^{2} \)
31 \( 1 - 5.48T + 31T^{2} \)
37 \( 1 + 8.35T + 37T^{2} \)
41 \( 1 - 4.43T + 41T^{2} \)
43 \( 1 - 0.890T + 43T^{2} \)
47 \( 1 + 5.98T + 47T^{2} \)
53 \( 1 + 3.04T + 53T^{2} \)
59 \( 1 - 14.5T + 59T^{2} \)
61 \( 1 - 0.262T + 61T^{2} \)
67 \( 1 + 9.92T + 67T^{2} \)
71 \( 1 - 14.4T + 71T^{2} \)
73 \( 1 - 6.46T + 73T^{2} \)
79 \( 1 + 6.44T + 79T^{2} \)
83 \( 1 - 13.8T + 83T^{2} \)
89 \( 1 - 6.07T + 89T^{2} \)
97 \( 1 + 9.15T + 97T^{2} \)
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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

−8.570315706786230204763785649959, −8.040103489708942037553867019680, −6.71107378256092318480917154091, −6.31582026540283134708967845712, −5.50284793320222838141876405299, −4.96969383370784598533531362607, −3.81635880675668823828951283942, −2.80537046324061186864941574385, −1.64221486042685238468882208153, −0.898501735722457882071198475256, 0.898501735722457882071198475256, 1.64221486042685238468882208153, 2.80537046324061186864941574385, 3.81635880675668823828951283942, 4.96969383370784598533531362607, 5.50284793320222838141876405299, 6.31582026540283134708967845712, 6.71107378256092318480917154091, 8.040103489708942037553867019680, 8.570315706786230204763785649959

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