Properties

Label 2-4004-1.1-c1-0-11
Degree $2$
Conductor $4004$
Sign $1$
Analytic cond. $31.9721$
Root an. cond. $5.65438$
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  + 0.0675·3-s − 1.58·5-s + 7-s − 2.99·9-s + 11-s − 13-s − 0.107·15-s − 7.43·17-s − 0.425·19-s + 0.0675·21-s + 2.83·23-s − 2.47·25-s − 0.405·27-s + 2.09·29-s + 9.18·31-s + 0.0675·33-s − 1.58·35-s + 3.75·37-s − 0.0675·39-s − 3.36·41-s − 7.73·43-s + 4.75·45-s + 9.56·47-s + 49-s − 0.502·51-s − 3.70·53-s − 1.58·55-s + ⋯
L(s)  = 1  + 0.0390·3-s − 0.710·5-s + 0.377·7-s − 0.998·9-s + 0.301·11-s − 0.277·13-s − 0.0277·15-s − 1.80·17-s − 0.0976·19-s + 0.0147·21-s + 0.591·23-s − 0.495·25-s − 0.0779·27-s + 0.389·29-s + 1.64·31-s + 0.0117·33-s − 0.268·35-s + 0.617·37-s − 0.0108·39-s − 0.526·41-s − 1.17·43-s + 0.709·45-s + 1.39·47-s + 0.142·49-s − 0.0703·51-s − 0.508·53-s − 0.214·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(4004\)    =    \(2^{2} \cdot 7 \cdot 11 \cdot 13\)
Sign: $1$
Analytic conductor: \(31.9721\)
Root analytic conductor: \(5.65438\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 4004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.219337877\)
\(L(\frac12)\) \(\approx\) \(1.219337877\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
7 \( 1 - T \)
11 \( 1 - T \)
13 \( 1 + T \)
good3 \( 1 - 0.0675T + 3T^{2} \)
5 \( 1 + 1.58T + 5T^{2} \)
17 \( 1 + 7.43T + 17T^{2} \)
19 \( 1 + 0.425T + 19T^{2} \)
23 \( 1 - 2.83T + 23T^{2} \)
29 \( 1 - 2.09T + 29T^{2} \)
31 \( 1 - 9.18T + 31T^{2} \)
37 \( 1 - 3.75T + 37T^{2} \)
41 \( 1 + 3.36T + 41T^{2} \)
43 \( 1 + 7.73T + 43T^{2} \)
47 \( 1 - 9.56T + 47T^{2} \)
53 \( 1 + 3.70T + 53T^{2} \)
59 \( 1 - 14.9T + 59T^{2} \)
61 \( 1 - 1.17T + 61T^{2} \)
67 \( 1 + 0.996T + 67T^{2} \)
71 \( 1 - 6.57T + 71T^{2} \)
73 \( 1 + 4.05T + 73T^{2} \)
79 \( 1 - 8.23T + 79T^{2} \)
83 \( 1 - 17.0T + 83T^{2} \)
89 \( 1 + 10.8T + 89T^{2} \)
97 \( 1 - 11.5T + 97T^{2} \)
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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

−8.401259616060922584077962217392, −7.919791430353267366392802203365, −6.88059793859993284321943077009, −6.41611558349575755296717659148, −5.37903832373222249631241093256, −4.60702767797411899295015015292, −3.94884615160668274527250826973, −2.89796323006007384461581430595, −2.12211351824731037292838855785, −0.60909132930974332063683414057, 0.60909132930974332063683414057, 2.12211351824731037292838855785, 2.89796323006007384461581430595, 3.94884615160668274527250826973, 4.60702767797411899295015015292, 5.37903832373222249631241093256, 6.41611558349575755296717659148, 6.88059793859993284321943077009, 7.919791430353267366392802203365, 8.401259616060922584077962217392

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