Properties

Label 2-4002-1.1-c1-0-37
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 − 1.59·5-s + 6-s + 1.20·7-s + 8-s + 9-s − 1.59·10-s + 4.80·11-s + 12-s + 1.05·13-s + 1.20·14-s − 1.59·15-s + 16-s − 2.39·17-s + 18-s + 6.34·19-s − 1.59·20-s + 1.20·21-s + 4.80·22-s + 23-s + 24-s − 2.46·25-s + 1.05·26-s + 27-s + 1.20·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 0.711·5-s + 0.408·6-s + 0.456·7-s + 0.353·8-s + 0.333·9-s − 0.503·10-s + 1.44·11-s + 0.288·12-s + 0.292·13-s + 0.322·14-s − 0.410·15-s + 0.250·16-s − 0.580·17-s + 0.235·18-s + 1.45·19-s − 0.355·20-s + 0.263·21-s + 1.02·22-s + 0.208·23-s + 0.204·24-s − 0.493·25-s + 0.206·26-s + 0.192·27-s + 0.228·28-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

\(L(1)\) \(\approx\) \(4.051980884\)
\(L(\frac12)\) \(\approx\) \(4.051980884\)
\(L(\frac{3}{2})\) 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 \( 1 + 1.59T + 5T^{2} \)
7 \( 1 - 1.20T + 7T^{2} \)
11 \( 1 - 4.80T + 11T^{2} \)
13 \( 1 - 1.05T + 13T^{2} \)
17 \( 1 + 2.39T + 17T^{2} \)
19 \( 1 - 6.34T + 19T^{2} \)
31 \( 1 + 5.20T + 31T^{2} \)
37 \( 1 - 7.70T + 37T^{2} \)
41 \( 1 - 1.69T + 41T^{2} \)
43 \( 1 + 7.44T + 43T^{2} \)
47 \( 1 - 8.28T + 47T^{2} \)
53 \( 1 - 1.92T + 53T^{2} \)
59 \( 1 - 2.23T + 59T^{2} \)
61 \( 1 + 5.98T + 61T^{2} \)
67 \( 1 - 1.48T + 67T^{2} \)
71 \( 1 - 7.48T + 71T^{2} \)
73 \( 1 + 3.40T + 73T^{2} \)
79 \( 1 - 17.0T + 79T^{2} \)
83 \( 1 - 4.46T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 17.6T + 97T^{2} \)
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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

−8.297665787189093524887443649523, −7.66754123615212998188160495591, −7.02807550578153103591826098208, −6.27985531818846077060056030511, −5.35262137312306090950320880047, −4.45931099500903418374435712032, −3.82786914155815925886099028636, −3.26204302429088411895816158371, −2.07704248846980944824050433434, −1.09600211832992946758107468033, 1.09600211832992946758107468033, 2.07704248846980944824050433434, 3.26204302429088411895816158371, 3.82786914155815925886099028636, 4.45931099500903418374435712032, 5.35262137312306090950320880047, 6.27985531818846077060056030511, 7.02807550578153103591826098208, 7.66754123615212998188160495591, 8.297665787189093524887443649523

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