Properties

Label 2-4002-1.1-c1-0-41
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 + 3.32·5-s + 6-s + 5.20·7-s − 8-s + 9-s − 3.32·10-s + 3.32·11-s − 12-s − 1.32·13-s − 5.20·14-s − 3.32·15-s + 16-s + 4·17-s − 18-s − 1.20·19-s + 3.32·20-s − 5.20·21-s − 3.32·22-s + 23-s + 24-s + 6.07·25-s + 1.32·26-s − 27-s + 5.20·28-s + ⋯
L(s)  = 1  − 0.707·2-s − 0.577·3-s + 0.5·4-s + 1.48·5-s + 0.408·6-s + 1.96·7-s − 0.353·8-s + 0.333·9-s − 1.05·10-s + 1.00·11-s − 0.288·12-s − 0.368·13-s − 1.39·14-s − 0.859·15-s + 0.250·16-s + 0.970·17-s − 0.235·18-s − 0.275·19-s + 0.744·20-s − 1.13·21-s − 0.709·22-s + 0.208·23-s + 0.204·24-s + 1.21·25-s + 0.260·26-s − 0.192·27-s + 0.983·28-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

\(L(1)\) \(\approx\) \(2.244021788\)
\(L(\frac12)\) \(\approx\) \(2.244021788\)
\(L(\frac{3}{2})\) 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 \( 1 - 3.32T + 5T^{2} \)
7 \( 1 - 5.20T + 7T^{2} \)
11 \( 1 - 3.32T + 11T^{2} \)
13 \( 1 + 1.32T + 13T^{2} \)
17 \( 1 - 4T + 17T^{2} \)
19 \( 1 + 1.20T + 19T^{2} \)
31 \( 1 + 1.32T + 31T^{2} \)
37 \( 1 + 5.07T + 37T^{2} \)
41 \( 1 + 0.924T + 41T^{2} \)
43 \( 1 - 11.8T + 43T^{2} \)
47 \( 1 + 10.6T + 47T^{2} \)
53 \( 1 - 6.65T + 53T^{2} \)
59 \( 1 - 11.3T + 59T^{2} \)
61 \( 1 - 1.32T + 61T^{2} \)
67 \( 1 - 1.32T + 67T^{2} \)
71 \( 1 + 9.58T + 71T^{2} \)
73 \( 1 + 2.25T + 73T^{2} \)
79 \( 1 + 8.40T + 79T^{2} \)
83 \( 1 + 9.60T + 83T^{2} \)
89 \( 1 + 10.6T + 89T^{2} \)
97 \( 1 - 13.4T + 97T^{2} \)
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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

−8.647132380316788731141949980551, −7.68100193527988351393627877972, −7.08479577864685174762188385256, −6.17549682787975760287567984057, −5.51089934227139874907993617740, −4.99473806584577512653746352047, −3.98432218213623488561902477217, −2.46490656574028925716209401381, −1.64046058496129349032308733341, −1.14681366607036368655386826491, 1.14681366607036368655386826491, 1.64046058496129349032308733341, 2.46490656574028925716209401381, 3.98432218213623488561902477217, 4.99473806584577512653746352047, 5.51089934227139874907993617740, 6.17549682787975760287567984057, 7.08479577864685174762188385256, 7.68100193527988351393627877972, 8.647132380316788731141949980551

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