Properties

Label 2-8004-1.1-c1-0-81
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 $1$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 2.48·5-s − 0.0448·7-s + 9-s − 0.683·11-s − 6.72·13-s − 2.48·15-s − 1.19·17-s + 1.53·19-s + 0.0448·21-s + 23-s + 1.15·25-s − 27-s − 29-s + 1.40·31-s + 0.683·33-s − 0.111·35-s + 8.68·37-s + 6.72·39-s + 11.5·41-s − 6.28·43-s + 2.48·45-s + 5.00·47-s − 6.99·49-s + 1.19·51-s − 1.85·53-s − 1.69·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.10·5-s − 0.0169·7-s + 0.333·9-s − 0.206·11-s − 1.86·13-s − 0.640·15-s − 0.289·17-s + 0.351·19-s + 0.00978·21-s + 0.208·23-s + 0.231·25-s − 0.192·27-s − 0.185·29-s + 0.252·31-s + 0.118·33-s − 0.0188·35-s + 1.42·37-s + 1.07·39-s + 1.80·41-s − 0.958·43-s + 0.369·45-s + 0.729·47-s − 0.999·49-s + 0.167·51-s − 0.254·53-s − 0.228·55-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: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(1\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ -1)\)

Particular Values

\(L(1)\) \(=\) \(0\)
\(L(\frac12)\) \(=\) \(0\)
\(L(\frac{3}{2})\) 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 \( 1 - 2.48T + 5T^{2} \)
7 \( 1 + 0.0448T + 7T^{2} \)
11 \( 1 + 0.683T + 11T^{2} \)
13 \( 1 + 6.72T + 13T^{2} \)
17 \( 1 + 1.19T + 17T^{2} \)
19 \( 1 - 1.53T + 19T^{2} \)
31 \( 1 - 1.40T + 31T^{2} \)
37 \( 1 - 8.68T + 37T^{2} \)
41 \( 1 - 11.5T + 41T^{2} \)
43 \( 1 + 6.28T + 43T^{2} \)
47 \( 1 - 5.00T + 47T^{2} \)
53 \( 1 + 1.85T + 53T^{2} \)
59 \( 1 + 2.14T + 59T^{2} \)
61 \( 1 + 2.75T + 61T^{2} \)
67 \( 1 + 0.0568T + 67T^{2} \)
71 \( 1 + 1.36T + 71T^{2} \)
73 \( 1 + 1.19T + 73T^{2} \)
79 \( 1 - 0.0825T + 79T^{2} \)
83 \( 1 - 7.83T + 83T^{2} \)
89 \( 1 + 0.473T + 89T^{2} \)
97 \( 1 + 5.20T + 97T^{2} \)
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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

−7.48474226584635283252094980760, −6.66922460352753795612891370882, −6.09139946335339628711134058798, −5.36545967214146839660792552090, −4.88016498295292940436411707479, −4.11531450904470545288277800006, −2.78646473486968306180134515911, −2.31549568410892928583216848976, −1.26991776593545882670296912952, 0, 1.26991776593545882670296912952, 2.31549568410892928583216848976, 2.78646473486968306180134515911, 4.11531450904470545288277800006, 4.88016498295292940436411707479, 5.36545967214146839660792552090, 6.09139946335339628711134058798, 6.66922460352753795612891370882, 7.48474226584635283252094980760

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