Properties

Label 2-4010-1.1-c1-0-127
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s − 0.375·3-s + 4-s + 5-s − 0.375·6-s + 3.43·7-s + 8-s − 2.85·9-s + 10-s − 3.38·11-s − 0.375·12-s − 5.75·13-s + 3.43·14-s − 0.375·15-s + 16-s − 0.0635·17-s − 2.85·18-s − 1.00·19-s + 20-s − 1.28·21-s − 3.38·22-s − 3.87·23-s − 0.375·24-s + 25-s − 5.75·26-s + 2.19·27-s + 3.43·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.216·3-s + 0.5·4-s + 0.447·5-s − 0.153·6-s + 1.29·7-s + 0.353·8-s − 0.953·9-s + 0.316·10-s − 1.02·11-s − 0.108·12-s − 1.59·13-s + 0.918·14-s − 0.0968·15-s + 0.250·16-s − 0.0154·17-s − 0.673·18-s − 0.231·19-s + 0.223·20-s − 0.281·21-s − 0.721·22-s − 0.808·23-s − 0.0765·24-s + 0.200·25-s − 1.12·26-s + 0.423·27-s + 0.649·28-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: \(1\)
Selberg data: \((2,\ 4010,\ (\ :1/2),\ -1)\)

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)$
bad2 \( 1 - T \)
5 \( 1 - T \)
401 \( 1 + T \)
good3 \( 1 + 0.375T + 3T^{2} \)
7 \( 1 - 3.43T + 7T^{2} \)
11 \( 1 + 3.38T + 11T^{2} \)
13 \( 1 + 5.75T + 13T^{2} \)
17 \( 1 + 0.0635T + 17T^{2} \)
19 \( 1 + 1.00T + 19T^{2} \)
23 \( 1 + 3.87T + 23T^{2} \)
29 \( 1 - 2.35T + 29T^{2} \)
31 \( 1 + 6.49T + 31T^{2} \)
37 \( 1 + 5.12T + 37T^{2} \)
41 \( 1 + 10.6T + 41T^{2} \)
43 \( 1 - 1.98T + 43T^{2} \)
47 \( 1 + 11.7T + 47T^{2} \)
53 \( 1 - 0.644T + 53T^{2} \)
59 \( 1 - 7.31T + 59T^{2} \)
61 \( 1 + 3.20T + 61T^{2} \)
67 \( 1 + 2.21T + 67T^{2} \)
71 \( 1 - 5.30T + 71T^{2} \)
73 \( 1 - 9.80T + 73T^{2} \)
79 \( 1 + 14.8T + 79T^{2} \)
83 \( 1 - 0.940T + 83T^{2} \)
89 \( 1 - 14.4T + 89T^{2} \)
97 \( 1 + 2.27T + 97T^{2} \)
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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

−8.043855850233507402915108019423, −7.33569156674108602990265557868, −6.50617956494430731064101353299, −5.48015198379905790114079285935, −5.16878163874490625255096566892, −4.61540020948236489740475882547, −3.35879912011386665408072376316, −2.40499120838037847733085599115, −1.83063766644060458114333581584, 0, 1.83063766644060458114333581584, 2.40499120838037847733085599115, 3.35879912011386665408072376316, 4.61540020948236489740475882547, 5.16878163874490625255096566892, 5.48015198379905790114079285935, 6.50617956494430731064101353299, 7.33569156674108602990265557868, 8.043855850233507402915108019423

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