Properties

Label 2-4002-1.1-c1-0-26
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 + 3.65·5-s − 6-s − 4.08·7-s + 8-s + 9-s + 3.65·10-s − 2.11·11-s − 12-s + 2.59·13-s − 4.08·14-s − 3.65·15-s + 16-s + 1.54·17-s + 18-s + 4.60·19-s + 3.65·20-s + 4.08·21-s − 2.11·22-s − 23-s − 24-s + 8.37·25-s + 2.59·26-s − 27-s − 4.08·28-s + ⋯
L(s)  = 1  + 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.63·5-s − 0.408·6-s − 1.54·7-s + 0.353·8-s + 0.333·9-s + 1.15·10-s − 0.636·11-s − 0.288·12-s + 0.719·13-s − 1.09·14-s − 0.944·15-s + 0.250·16-s + 0.375·17-s + 0.235·18-s + 1.05·19-s + 0.817·20-s + 0.890·21-s − 0.449·22-s − 0.208·23-s − 0.204·24-s + 1.67·25-s + 0.508·26-s − 0.192·27-s − 0.771·28-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

\(L(1)\) \(\approx\) \(2.994852180\)
\(L(\frac12)\) \(\approx\) \(2.994852180\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.65T + 5T^{2} \)
7 \( 1 + 4.08T + 7T^{2} \)
11 \( 1 + 2.11T + 11T^{2} \)
13 \( 1 - 2.59T + 13T^{2} \)
17 \( 1 - 1.54T + 17T^{2} \)
19 \( 1 - 4.60T + 19T^{2} \)
31 \( 1 - 0.201T + 31T^{2} \)
37 \( 1 - 4.67T + 37T^{2} \)
41 \( 1 - 1.23T + 41T^{2} \)
43 \( 1 + 3.50T + 43T^{2} \)
47 \( 1 + 2.84T + 47T^{2} \)
53 \( 1 - 7.19T + 53T^{2} \)
59 \( 1 - 9.32T + 59T^{2} \)
61 \( 1 - 4.11T + 61T^{2} \)
67 \( 1 - 1.70T + 67T^{2} \)
71 \( 1 - 4.12T + 71T^{2} \)
73 \( 1 - 4.28T + 73T^{2} \)
79 \( 1 + 5.51T + 79T^{2} \)
83 \( 1 + 0.151T + 83T^{2} \)
89 \( 1 - 8.36T + 89T^{2} \)
97 \( 1 + 1.09T + 97T^{2} \)
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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

−8.525884678838935585765125896760, −7.34887392559514280117882047623, −6.66172696758325241560775003424, −6.02980401919331507715883655168, −5.65643835417496988725574523847, −4.98871716874039502037053751048, −3.74723102192508309599025427954, −2.99594556506652178256418450335, −2.15088068150512236490183329351, −0.933086844960607108684809475407, 0.933086844960607108684809475407, 2.15088068150512236490183329351, 2.99594556506652178256418450335, 3.74723102192508309599025427954, 4.98871716874039502037053751048, 5.65643835417496988725574523847, 6.02980401919331507715883655168, 6.66172696758325241560775003424, 7.34887392559514280117882047623, 8.525884678838935585765125896760

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