Properties

Label 2-4010-1.1-c1-0-86
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 + 1.74·3-s + 4-s − 5-s + 1.74·6-s + 3.19·7-s + 8-s + 0.0421·9-s − 10-s + 2.45·11-s + 1.74·12-s + 5.45·13-s + 3.19·14-s − 1.74·15-s + 16-s − 2.49·17-s + 0.0421·18-s + 2.96·19-s − 20-s + 5.58·21-s + 2.45·22-s − 1.55·23-s + 1.74·24-s + 25-s + 5.45·26-s − 5.15·27-s + 3.19·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 1.00·3-s + 0.5·4-s − 0.447·5-s + 0.712·6-s + 1.20·7-s + 0.353·8-s + 0.0140·9-s − 0.316·10-s + 0.741·11-s + 0.503·12-s + 1.51·13-s + 0.855·14-s − 0.450·15-s + 0.250·16-s − 0.606·17-s + 0.00993·18-s + 0.680·19-s − 0.223·20-s + 1.21·21-s + 0.524·22-s − 0.324·23-s + 0.356·24-s + 0.200·25-s + 1.06·26-s − 0.992·27-s + 0.604·28-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

\(L(1)\) \(\approx\) \(5.060787318\)
\(L(\frac12)\) \(\approx\) \(5.060787318\)
\(L(\frac{3}{2})\) 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 \( 1 - 1.74T + 3T^{2} \)
7 \( 1 - 3.19T + 7T^{2} \)
11 \( 1 - 2.45T + 11T^{2} \)
13 \( 1 - 5.45T + 13T^{2} \)
17 \( 1 + 2.49T + 17T^{2} \)
19 \( 1 - 2.96T + 19T^{2} \)
23 \( 1 + 1.55T + 23T^{2} \)
29 \( 1 + 9.72T + 29T^{2} \)
31 \( 1 - 2.56T + 31T^{2} \)
37 \( 1 - 6.95T + 37T^{2} \)
41 \( 1 - 10.1T + 41T^{2} \)
43 \( 1 + 2.04T + 43T^{2} \)
47 \( 1 + 6.40T + 47T^{2} \)
53 \( 1 - 10.1T + 53T^{2} \)
59 \( 1 + 2.94T + 59T^{2} \)
61 \( 1 + 6.84T + 61T^{2} \)
67 \( 1 - 9.37T + 67T^{2} \)
71 \( 1 - 2.97T + 71T^{2} \)
73 \( 1 - 2.99T + 73T^{2} \)
79 \( 1 + 6.79T + 79T^{2} \)
83 \( 1 + 13.8T + 83T^{2} \)
89 \( 1 + 5.48T + 89T^{2} \)
97 \( 1 + 8.44T + 97T^{2} \)
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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

−8.301672610393036736919521708310, −7.86186661138549692685233713679, −7.10397132006299905946745266450, −6.10277857018206746552299456580, −5.47218894666315722811519873036, −4.32902419928444385625626106643, −3.93183633352688384785625689509, −3.12259974573650914514201740147, −2.10492131592638297921540065044, −1.24815080733874113189107892638, 1.24815080733874113189107892638, 2.10492131592638297921540065044, 3.12259974573650914514201740147, 3.93183633352688384785625689509, 4.32902419928444385625626106643, 5.47218894666315722811519873036, 6.10277857018206746552299456580, 7.10397132006299905946745266450, 7.86186661138549692685233713679, 8.301672610393036736919521708310

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