Properties

Label 2-4010-1.1-c1-0-37
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 + 1.51·3-s + 4-s − 5-s − 1.51·6-s + 1.25·7-s − 8-s − 0.692·9-s + 10-s − 4.69·11-s + 1.51·12-s + 3.82·13-s − 1.25·14-s − 1.51·15-s + 16-s − 0.345·17-s + 0.692·18-s + 5.84·19-s − 20-s + 1.91·21-s + 4.69·22-s + 4.93·23-s − 1.51·24-s + 25-s − 3.82·26-s − 5.60·27-s + 1.25·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 0.876·3-s + 0.5·4-s − 0.447·5-s − 0.620·6-s + 0.475·7-s − 0.353·8-s − 0.230·9-s + 0.316·10-s − 1.41·11-s + 0.438·12-s + 1.05·13-s − 0.336·14-s − 0.392·15-s + 0.250·16-s − 0.0837·17-s + 0.163·18-s + 1.34·19-s − 0.223·20-s + 0.417·21-s + 1.00·22-s + 1.02·23-s − 0.310·24-s + 0.200·25-s − 0.749·26-s − 1.07·27-s + 0.237·28-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

\(L(1)\) \(\approx\) \(1.671028182\)
\(L(\frac12)\) \(\approx\) \(1.671028182\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.51T + 3T^{2} \)
7 \( 1 - 1.25T + 7T^{2} \)
11 \( 1 + 4.69T + 11T^{2} \)
13 \( 1 - 3.82T + 13T^{2} \)
17 \( 1 + 0.345T + 17T^{2} \)
19 \( 1 - 5.84T + 19T^{2} \)
23 \( 1 - 4.93T + 23T^{2} \)
29 \( 1 - 7.80T + 29T^{2} \)
31 \( 1 + 1.46T + 31T^{2} \)
37 \( 1 - 0.209T + 37T^{2} \)
41 \( 1 + 0.370T + 41T^{2} \)
43 \( 1 + 8.41T + 43T^{2} \)
47 \( 1 - 4.61T + 47T^{2} \)
53 \( 1 + 10.3T + 53T^{2} \)
59 \( 1 - 0.967T + 59T^{2} \)
61 \( 1 - 6.90T + 61T^{2} \)
67 \( 1 + 10.0T + 67T^{2} \)
71 \( 1 - 6.14T + 71T^{2} \)
73 \( 1 - 6.53T + 73T^{2} \)
79 \( 1 - 15.3T + 79T^{2} \)
83 \( 1 - 7.49T + 83T^{2} \)
89 \( 1 + 12.6T + 89T^{2} \)
97 \( 1 - 17.7T + 97T^{2} \)
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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

−8.343304169925324294691307148631, −7.969119473723103428698986248460, −7.35555454049288050902565212174, −6.42890110184195560533034079266, −5.42233609237396752920717021417, −4.75381227905781501776489603965, −3.36335688671794989568292381502, −3.04042209616799358358554112340, −1.98035293513221766604615293711, −0.789707506582053095966475083833, 0.789707506582053095966475083833, 1.98035293513221766604615293711, 3.04042209616799358358554112340, 3.36335688671794989568292381502, 4.75381227905781501776489603965, 5.42233609237396752920717021417, 6.42890110184195560533034079266, 7.35555454049288050902565212174, 7.969119473723103428698986248460, 8.343304169925324294691307148631

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