Properties

Label 2-4002-1.1-c1-0-12
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 + 0.230·5-s + 6-s + 4.59·7-s − 8-s + 9-s − 0.230·10-s − 4.50·11-s − 12-s + 1.15·13-s − 4.59·14-s − 0.230·15-s + 16-s − 7.42·17-s − 18-s − 5.94·19-s + 0.230·20-s − 4.59·21-s + 4.50·22-s + 23-s + 24-s − 4.94·25-s − 1.15·26-s − 27-s + 4.59·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 0.102·5-s + 0.408·6-s + 1.73·7-s − 0.353·8-s + 0.333·9-s − 0.0728·10-s − 1.35·11-s − 0.288·12-s + 0.320·13-s − 1.22·14-s − 0.0594·15-s + 0.250·16-s − 1.80·17-s − 0.235·18-s − 1.36·19-s + 0.0514·20-s − 1.00·21-s + 0.960·22-s + 0.208·23-s + 0.204·24-s − 0.989·25-s − 0.226·26-s − 0.192·27-s + 0.868·28-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

\(L(1)\) \(\approx\) \(1.048099366\)
\(L(\frac12)\) \(\approx\) \(1.048099366\)
\(L(\frac{3}{2})\) 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 \( 1 - 0.230T + 5T^{2} \)
7 \( 1 - 4.59T + 7T^{2} \)
11 \( 1 + 4.50T + 11T^{2} \)
13 \( 1 - 1.15T + 13T^{2} \)
17 \( 1 + 7.42T + 17T^{2} \)
19 \( 1 + 5.94T + 19T^{2} \)
31 \( 1 - 6.50T + 31T^{2} \)
37 \( 1 - 7.29T + 37T^{2} \)
41 \( 1 - 0.0953T + 41T^{2} \)
43 \( 1 - 0.941T + 43T^{2} \)
47 \( 1 + 0.598T + 47T^{2} \)
53 \( 1 - 13.3T + 53T^{2} \)
59 \( 1 + 2.94T + 59T^{2} \)
61 \( 1 - 10.8T + 61T^{2} \)
67 \( 1 - 2.09T + 67T^{2} \)
71 \( 1 - 10.9T + 71T^{2} \)
73 \( 1 + 12.3T + 73T^{2} \)
79 \( 1 - 16.5T + 79T^{2} \)
83 \( 1 + 1.71T + 83T^{2} \)
89 \( 1 - 12.5T + 89T^{2} \)
97 \( 1 + 1.82T + 97T^{2} \)
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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

−8.308141225689569048871619558886, −7.958710481808238266256809122228, −7.09062906218052413076981248971, −6.28636572405820789937375183650, −5.50863859880605912001720531561, −4.70674940525600118228113621584, −4.14820613937935349855855121250, −2.38882215644980644418351265159, −2.01502943840024056540401026618, −0.65810877460570271866895200525, 0.65810877460570271866895200525, 2.01502943840024056540401026618, 2.38882215644980644418351265159, 4.14820613937935349855855121250, 4.70674940525600118228113621584, 5.50863859880605912001720531561, 6.28636572405820789937375183650, 7.09062906218052413076981248971, 7.958710481808238266256809122228, 8.308141225689569048871619558886

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