Properties

Label 2-4002-1.1-c1-0-11
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 − 4.09·5-s + 6-s − 2.85·7-s + 8-s + 9-s − 4.09·10-s + 0.493·11-s + 12-s − 5.03·13-s − 2.85·14-s − 4.09·15-s + 16-s − 0.585·17-s + 18-s + 0.996·19-s − 4.09·20-s − 2.85·21-s + 0.493·22-s + 23-s + 24-s + 11.7·25-s − 5.03·26-s + 27-s − 2.85·28-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.577·3-s + 0.5·4-s − 1.82·5-s + 0.408·6-s − 1.07·7-s + 0.353·8-s + 0.333·9-s − 1.29·10-s + 0.148·11-s + 0.288·12-s − 1.39·13-s − 0.762·14-s − 1.05·15-s + 0.250·16-s − 0.141·17-s + 0.235·18-s + 0.228·19-s − 0.914·20-s − 0.622·21-s + 0.105·22-s + 0.208·23-s + 0.204·24-s + 2.34·25-s − 0.987·26-s + 0.192·27-s − 0.539·28-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

\(L(1)\) \(\approx\) \(1.833068279\)
\(L(\frac12)\) \(\approx\) \(1.833068279\)
\(L(\frac{3}{2})\) 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 \( 1 + 4.09T + 5T^{2} \)
7 \( 1 + 2.85T + 7T^{2} \)
11 \( 1 - 0.493T + 11T^{2} \)
13 \( 1 + 5.03T + 13T^{2} \)
17 \( 1 + 0.585T + 17T^{2} \)
19 \( 1 - 0.996T + 19T^{2} \)
31 \( 1 - 4.90T + 31T^{2} \)
37 \( 1 + 1.11T + 37T^{2} \)
41 \( 1 - 7.29T + 41T^{2} \)
43 \( 1 - 6.49T + 43T^{2} \)
47 \( 1 - 4.69T + 47T^{2} \)
53 \( 1 - 7.44T + 53T^{2} \)
59 \( 1 - 1.14T + 59T^{2} \)
61 \( 1 + 6.67T + 61T^{2} \)
67 \( 1 - 16.1T + 67T^{2} \)
71 \( 1 - 6.89T + 71T^{2} \)
73 \( 1 - 1.12T + 73T^{2} \)
79 \( 1 + 10.1T + 79T^{2} \)
83 \( 1 + 0.118T + 83T^{2} \)
89 \( 1 + 12.2T + 89T^{2} \)
97 \( 1 - 1.37T + 97T^{2} \)
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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

−8.271007537483241312296107881368, −7.50850245948629598537655109300, −7.16469334894771380670589032414, −6.41376255720479517770213056365, −5.25855755585126495031644283423, −4.37663894612099892163713793325, −3.89938688455993292908169592663, −3.07649981055823145114303463744, −2.49966754385172427101149416578, −0.64937566553249103465873210922, 0.64937566553249103465873210922, 2.49966754385172427101149416578, 3.07649981055823145114303463744, 3.89938688455993292908169592663, 4.37663894612099892163713793325, 5.25855755585126495031644283423, 6.41376255720479517770213056365, 7.16469334894771380670589032414, 7.50850245948629598537655109300, 8.271007537483241312296107881368

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