Properties

Label 2-4002-1.1-c1-0-30
Degree $2$
Conductor $4002$
Sign $1$
Analytic cond. $31.9561$
Root an. cond. $5.65297$
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 + 3-s + 4-s − 3.71·5-s + 6-s + 2.42·7-s + 8-s + 9-s − 3.71·10-s + 5.28·11-s + 12-s + 1.77·13-s + 2.42·14-s − 3.71·15-s + 16-s + 2.42·17-s + 18-s − 2.42·19-s − 3.71·20-s + 2.42·21-s + 5.28·22-s − 23-s + 24-s + 8.76·25-s + 1.77·26-s + 27-s + 2.42·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.65·5-s + 0.408·6-s + 0.916·7-s + 0.353·8-s + 0.333·9-s − 1.17·10-s + 1.59·11-s + 0.288·12-s + 0.492·13-s + 0.647·14-s − 0.958·15-s + 0.250·16-s + 0.588·17-s + 0.235·18-s − 0.556·19-s − 0.829·20-s + 0.529·21-s + 1.12·22-s − 0.208·23-s + 0.204·24-s + 1.75·25-s + 0.348·26-s + 0.192·27-s + 0.458·28-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4002 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4002\)    =    \(2 \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(31.9561\)
Root analytic conductor: \(5.65297\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4002,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(3.586166020\)
\(L(\frac12)\) \(\approx\) \(3.586166020\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 - T \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 3.71T + 5T^{2} \)
7 \( 1 - 2.42T + 7T^{2} \)
11 \( 1 - 5.28T + 11T^{2} \)
13 \( 1 - 1.77T + 13T^{2} \)
17 \( 1 - 2.42T + 17T^{2} \)
19 \( 1 + 2.42T + 19T^{2} \)
31 \( 1 + 7.19T + 31T^{2} \)
37 \( 1 + 8.82T + 37T^{2} \)
41 \( 1 - 11.7T + 41T^{2} \)
43 \( 1 - 4.45T + 43T^{2} \)
47 \( 1 + 10.5T + 47T^{2} \)
53 \( 1 - 8.75T + 53T^{2} \)
59 \( 1 - 12.3T + 59T^{2} \)
61 \( 1 - 8.37T + 61T^{2} \)
67 \( 1 - 5.55T + 67T^{2} \)
71 \( 1 + 1.49T + 71T^{2} \)
73 \( 1 + 4.66T + 73T^{2} \)
79 \( 1 - 6.41T + 79T^{2} \)
83 \( 1 + 1.75T + 83T^{2} \)
89 \( 1 + 4.75T + 89T^{2} \)
97 \( 1 - 10.1T + 97T^{2} \)
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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

−8.510052115793032859100934031598, −7.60022360754062212537696992463, −7.16264790735319393075443957970, −6.32642032177584553983700886712, −5.25748536513184171561118047583, −4.35280701315043342251846956946, −3.84923644699699218054776427774, −3.41425895700352252283425616021, −2.05540283793924472189019933892, −1.01791186376631129631626723294, 1.01791186376631129631626723294, 2.05540283793924472189019933892, 3.41425895700352252283425616021, 3.84923644699699218054776427774, 4.35280701315043342251846956946, 5.25748536513184171561118047583, 6.32642032177584553983700886712, 7.16264790735319393075443957970, 7.60022360754062212537696992463, 8.510052115793032859100934031598

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