Properties

Label 2-4010-1.1-c1-0-43
Degree $2$
Conductor $4010$
Sign $1$
Analytic cond. $32.0200$
Root an. cond. $5.65862$
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 − 2.99·3-s + 4-s + 5-s + 2.99·6-s + 0.839·7-s − 8-s + 5.95·9-s − 10-s + 4.32·11-s − 2.99·12-s + 0.593·13-s − 0.839·14-s − 2.99·15-s + 16-s − 0.302·17-s − 5.95·18-s + 3.41·19-s + 20-s − 2.51·21-s − 4.32·22-s + 1.14·23-s + 2.99·24-s + 25-s − 0.593·26-s − 8.84·27-s + 0.839·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 1.72·3-s + 0.5·4-s + 0.447·5-s + 1.22·6-s + 0.317·7-s − 0.353·8-s + 1.98·9-s − 0.316·10-s + 1.30·11-s − 0.863·12-s + 0.164·13-s − 0.224·14-s − 0.772·15-s + 0.250·16-s − 0.0734·17-s − 1.40·18-s + 0.784·19-s + 0.223·20-s − 0.547·21-s − 0.922·22-s + 0.238·23-s + 0.610·24-s + 0.200·25-s − 0.116·26-s − 1.70·27-s + 0.158·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4010 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4010\)    =    \(2 \cdot 5 \cdot 401\)
Sign: $1$
Analytic conductor: \(32.0200\)
Root analytic conductor: \(5.65862\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4010,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.084815687\)
\(L(\frac12)\) \(\approx\) \(1.084815687\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 + T \)
5 \( 1 - T \)
401 \( 1 + T \)
good3 \( 1 + 2.99T + 3T^{2} \)
7 \( 1 - 0.839T + 7T^{2} \)
11 \( 1 - 4.32T + 11T^{2} \)
13 \( 1 - 0.593T + 13T^{2} \)
17 \( 1 + 0.302T + 17T^{2} \)
19 \( 1 - 3.41T + 19T^{2} \)
23 \( 1 - 1.14T + 23T^{2} \)
29 \( 1 - 9.67T + 29T^{2} \)
31 \( 1 - 4.06T + 31T^{2} \)
37 \( 1 - 1.09T + 37T^{2} \)
41 \( 1 - 11.6T + 41T^{2} \)
43 \( 1 - 8.97T + 43T^{2} \)
47 \( 1 - 12.2T + 47T^{2} \)
53 \( 1 + 1.48T + 53T^{2} \)
59 \( 1 + 9.58T + 59T^{2} \)
61 \( 1 + 9.64T + 61T^{2} \)
67 \( 1 - 1.19T + 67T^{2} \)
71 \( 1 - 10.8T + 71T^{2} \)
73 \( 1 + 2.30T + 73T^{2} \)
79 \( 1 - 6.33T + 79T^{2} \)
83 \( 1 + 15.2T + 83T^{2} \)
89 \( 1 + 17.4T + 89T^{2} \)
97 \( 1 + 15.7T + 97T^{2} \)
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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

−8.525991773697084920864183597628, −7.51857657969818601470013584182, −6.86986536445362062670066956534, −6.16068874431270578679637765549, −5.81400051380636426285783380639, −4.77721590926887217575755919966, −4.16068100247615824528227042371, −2.73351542622297081367634031876, −1.33833354440766041077971234667, −0.865040811756209858316726757438, 0.865040811756209858316726757438, 1.33833354440766041077971234667, 2.73351542622297081367634031876, 4.16068100247615824528227042371, 4.77721590926887217575755919966, 5.81400051380636426285783380639, 6.16068874431270578679637765549, 6.86986536445362062670066956534, 7.51857657969818601470013584182, 8.525991773697084920864183597628

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