Properties

Label 2-4010-1.1-c1-0-77
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 + 2.32·3-s + 4-s + 5-s − 2.32·6-s + 1.00·7-s − 8-s + 2.39·9-s − 10-s − 0.100·11-s + 2.32·12-s + 4.75·13-s − 1.00·14-s + 2.32·15-s + 16-s + 7.59·17-s − 2.39·18-s + 6.05·19-s + 20-s + 2.32·21-s + 0.100·22-s − 5.89·23-s − 2.32·24-s + 25-s − 4.75·26-s − 1.41·27-s + 1.00·28-s + ⋯
L(s)  = 1  − 0.707·2-s + 1.34·3-s + 0.5·4-s + 0.447·5-s − 0.947·6-s + 0.378·7-s − 0.353·8-s + 0.797·9-s − 0.316·10-s − 0.0302·11-s + 0.670·12-s + 1.31·13-s − 0.267·14-s + 0.599·15-s + 0.250·16-s + 1.84·17-s − 0.563·18-s + 1.38·19-s + 0.223·20-s + 0.507·21-s + 0.0214·22-s − 1.22·23-s − 0.473·24-s + 0.200·25-s − 0.932·26-s − 0.271·27-s + 0.189·28-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

\(L(1)\) \(\approx\) \(3.006376949\)
\(L(\frac12)\) \(\approx\) \(3.006376949\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.32T + 3T^{2} \)
7 \( 1 - 1.00T + 7T^{2} \)
11 \( 1 + 0.100T + 11T^{2} \)
13 \( 1 - 4.75T + 13T^{2} \)
17 \( 1 - 7.59T + 17T^{2} \)
19 \( 1 - 6.05T + 19T^{2} \)
23 \( 1 + 5.89T + 23T^{2} \)
29 \( 1 + 3.17T + 29T^{2} \)
31 \( 1 - 3.07T + 31T^{2} \)
37 \( 1 - 12.0T + 37T^{2} \)
41 \( 1 + 2.81T + 41T^{2} \)
43 \( 1 - 11.9T + 43T^{2} \)
47 \( 1 + 11.1T + 47T^{2} \)
53 \( 1 + 12.5T + 53T^{2} \)
59 \( 1 + 8.79T + 59T^{2} \)
61 \( 1 + 9.57T + 61T^{2} \)
67 \( 1 + 16.1T + 67T^{2} \)
71 \( 1 - 3.80T + 71T^{2} \)
73 \( 1 + 1.60T + 73T^{2} \)
79 \( 1 - 0.972T + 79T^{2} \)
83 \( 1 - 6.67T + 83T^{2} \)
89 \( 1 + 6.50T + 89T^{2} \)
97 \( 1 + 1.29T + 97T^{2} \)
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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

−8.314136854836832047330819499956, −7.82251160417501267655593712373, −7.56600079385447167444765229893, −6.13768004622665382696267136568, −5.80854984950684063988254455314, −4.53076007138765228035130835840, −3.35857692878697662971079111241, −3.05485066071692560409201500870, −1.81549217095831219392545195490, −1.16566686397888874461010076703, 1.16566686397888874461010076703, 1.81549217095831219392545195490, 3.05485066071692560409201500870, 3.35857692878697662971079111241, 4.53076007138765228035130835840, 5.80854984950684063988254455314, 6.13768004622665382696267136568, 7.56600079385447167444765229893, 7.82251160417501267655593712373, 8.314136854836832047330819499956

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