Properties

Label 2-4010-1.1-c1-0-105
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 + 2.70·3-s + 4-s + 5-s + 2.70·6-s + 0.508·7-s + 8-s + 4.34·9-s + 10-s − 0.524·11-s + 2.70·12-s + 6.69·13-s + 0.508·14-s + 2.70·15-s + 16-s − 3.24·17-s + 4.34·18-s + 0.166·19-s + 20-s + 1.37·21-s − 0.524·22-s + 1.07·23-s + 2.70·24-s + 25-s + 6.69·26-s + 3.63·27-s + 0.508·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.56·3-s + 0.5·4-s + 0.447·5-s + 1.10·6-s + 0.192·7-s + 0.353·8-s + 1.44·9-s + 0.316·10-s − 0.158·11-s + 0.782·12-s + 1.85·13-s + 0.136·14-s + 0.699·15-s + 0.250·16-s − 0.787·17-s + 1.02·18-s + 0.0381·19-s + 0.223·20-s + 0.300·21-s − 0.111·22-s + 0.223·23-s + 0.553·24-s + 0.200·25-s + 1.31·26-s + 0.699·27-s + 0.0961·28-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

\(L(1)\) \(\approx\) \(6.369256224\)
\(L(\frac12)\) \(\approx\) \(6.369256224\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.70T + 3T^{2} \)
7 \( 1 - 0.508T + 7T^{2} \)
11 \( 1 + 0.524T + 11T^{2} \)
13 \( 1 - 6.69T + 13T^{2} \)
17 \( 1 + 3.24T + 17T^{2} \)
19 \( 1 - 0.166T + 19T^{2} \)
23 \( 1 - 1.07T + 23T^{2} \)
29 \( 1 - 2.11T + 29T^{2} \)
31 \( 1 + 7.15T + 31T^{2} \)
37 \( 1 + 6.51T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 3.01T + 43T^{2} \)
47 \( 1 + 7.56T + 47T^{2} \)
53 \( 1 - 4.08T + 53T^{2} \)
59 \( 1 + 1.50T + 59T^{2} \)
61 \( 1 - 7.63T + 61T^{2} \)
67 \( 1 + 9.42T + 67T^{2} \)
71 \( 1 + 13.6T + 71T^{2} \)
73 \( 1 - 3.63T + 73T^{2} \)
79 \( 1 - 14.5T + 79T^{2} \)
83 \( 1 - 6.44T + 83T^{2} \)
89 \( 1 + 4.50T + 89T^{2} \)
97 \( 1 - 6.30T + 97T^{2} \)
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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

−8.499195018051288566610545749313, −7.83083969779918678128195871767, −6.96908215382445121181440954273, −6.26758202442051018120889338074, −5.43395250924615547989254225974, −4.43253487397081338485045750439, −3.67624109117347050228537789564, −3.10269738366139222159990381373, −2.14544223238125580914865306712, −1.43263791609823524805724063243, 1.43263791609823524805724063243, 2.14544223238125580914865306712, 3.10269738366139222159990381373, 3.67624109117347050228537789564, 4.43253487397081338485045750439, 5.43395250924615547989254225974, 6.26758202442051018120889338074, 6.96908215382445121181440954273, 7.83083969779918678128195871767, 8.499195018051288566610545749313

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