Properties

Label 2-8004-1.1-c1-0-2
Degree $2$
Conductor $8004$
Sign $1$
Analytic cond. $63.9122$
Root an. cond. $7.99451$
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  + 3-s − 3.70·5-s − 1.62·7-s + 9-s − 4.14·11-s − 5.66·13-s − 3.70·15-s + 0.274·17-s − 7.65·19-s − 1.62·21-s − 23-s + 8.71·25-s + 27-s + 29-s − 9.73·31-s − 4.14·33-s + 5.99·35-s − 1.32·37-s − 5.66·39-s − 8.38·41-s + 11.4·43-s − 3.70·45-s − 7.35·47-s − 4.37·49-s + 0.274·51-s − 0.0615·53-s + 15.3·55-s + ⋯
L(s)  = 1  + 0.577·3-s − 1.65·5-s − 0.612·7-s + 0.333·9-s − 1.25·11-s − 1.57·13-s − 0.956·15-s + 0.0666·17-s − 1.75·19-s − 0.353·21-s − 0.208·23-s + 1.74·25-s + 0.192·27-s + 0.185·29-s − 1.74·31-s − 0.721·33-s + 1.01·35-s − 0.218·37-s − 0.906·39-s − 1.30·41-s + 1.74·43-s − 0.552·45-s − 1.07·47-s − 0.625·49-s + 0.0384·51-s − 0.00845·53-s + 2.07·55-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]

Invariants

Degree: \(2\)
Conductor: \(8004\)    =    \(2^{2} \cdot 3 \cdot 23 \cdot 29\)
Sign: $1$
Analytic conductor: \(63.9122\)
Root analytic conductor: \(7.99451\)
Motivic weight: \(1\)
Rational: no
Arithmetic: yes
Character: Trivial
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(0.1149480375\)
\(L(\frac12)\) \(\approx\) \(0.1149480375\)
\(L(\frac{3}{2})\) not available
\(L(1)\) not available

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad2 \( 1 \)
3 \( 1 - T \)
23 \( 1 + T \)
29 \( 1 - T \)
good5 \( 1 + 3.70T + 5T^{2} \)
7 \( 1 + 1.62T + 7T^{2} \)
11 \( 1 + 4.14T + 11T^{2} \)
13 \( 1 + 5.66T + 13T^{2} \)
17 \( 1 - 0.274T + 17T^{2} \)
19 \( 1 + 7.65T + 19T^{2} \)
31 \( 1 + 9.73T + 31T^{2} \)
37 \( 1 + 1.32T + 37T^{2} \)
41 \( 1 + 8.38T + 41T^{2} \)
43 \( 1 - 11.4T + 43T^{2} \)
47 \( 1 + 7.35T + 47T^{2} \)
53 \( 1 + 0.0615T + 53T^{2} \)
59 \( 1 + 1.61T + 59T^{2} \)
61 \( 1 - 8.30T + 61T^{2} \)
67 \( 1 - 1.88T + 67T^{2} \)
71 \( 1 + 12.7T + 71T^{2} \)
73 \( 1 + 1.88T + 73T^{2} \)
79 \( 1 + 13.8T + 79T^{2} \)
83 \( 1 + 10.8T + 83T^{2} \)
89 \( 1 - 13.3T + 89T^{2} \)
97 \( 1 + 5.34T + 97T^{2} \)
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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

−7.80577017499107927215973935617, −7.28113596602380580738115156246, −6.80793299194211442008601587355, −5.67941227771329127500172755167, −4.77817341759307409270921303847, −4.27043687146300992672786918470, −3.45954000842375492754365968938, −2.80459146215001240769540596489, −2.02077341141627581970612110557, −0.14939959031164832794127129155, 0.14939959031164832794127129155, 2.02077341141627581970612110557, 2.80459146215001240769540596489, 3.45954000842375492754365968938, 4.27043687146300992672786918470, 4.77817341759307409270921303847, 5.67941227771329127500172755167, 6.80793299194211442008601587355, 7.28113596602380580738115156246, 7.80577017499107927215973935617

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