Properties

Degree $2$
Conductor $8004$
Sign $1$
Motivic weight $1$
Primitive yes
Self-dual yes
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s + 4.15·5-s − 4.14·7-s + 9-s + 0.229·11-s − 4.10·13-s − 4.15·15-s + 6.38·17-s − 2.89·19-s + 4.14·21-s + 23-s + 12.2·25-s − 27-s + 29-s + 7.43·31-s − 0.229·33-s − 17.2·35-s − 6.31·37-s + 4.10·39-s + 5.76·41-s + 0.993·43-s + 4.15·45-s − 6.88·47-s + 10.1·49-s − 6.38·51-s + 6.39·53-s + 0.951·55-s + ⋯
L(s)  = 1  − 0.577·3-s + 1.85·5-s − 1.56·7-s + 0.333·9-s + 0.0690·11-s − 1.13·13-s − 1.07·15-s + 1.54·17-s − 0.663·19-s + 0.904·21-s + 0.208·23-s + 2.45·25-s − 0.192·27-s + 0.185·29-s + 1.33·31-s − 0.0398·33-s − 2.91·35-s − 1.03·37-s + 0.657·39-s + 0.900·41-s + 0.151·43-s + 0.619·45-s − 1.00·47-s + 1.45·49-s − 0.894·51-s + 0.878·53-s + 0.128·55-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: $1$
Motivic weight: \(1\)
Character: $\chi_{8004} (1, \cdot )$
Primitive: yes
Self-dual: yes
Analytic rank: \(0\)
Selberg data: \((2,\ 8004,\ (\ :1/2),\ 1)\)

Particular Values

\(L(1)\) \(\approx\) \(1.856177594\)
\(L(\frac12)\) \(\approx\) \(1.856177594\)
\(L(\frac{3}{2})\) 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 \( 1 - 4.15T + 5T^{2} \)
7 \( 1 + 4.14T + 7T^{2} \)
11 \( 1 - 0.229T + 11T^{2} \)
13 \( 1 + 4.10T + 13T^{2} \)
17 \( 1 - 6.38T + 17T^{2} \)
19 \( 1 + 2.89T + 19T^{2} \)
31 \( 1 - 7.43T + 31T^{2} \)
37 \( 1 + 6.31T + 37T^{2} \)
41 \( 1 - 5.76T + 41T^{2} \)
43 \( 1 - 0.993T + 43T^{2} \)
47 \( 1 + 6.88T + 47T^{2} \)
53 \( 1 - 6.39T + 53T^{2} \)
59 \( 1 + 3.98T + 59T^{2} \)
61 \( 1 + 4.98T + 61T^{2} \)
67 \( 1 - 2.09T + 67T^{2} \)
71 \( 1 + 13.7T + 71T^{2} \)
73 \( 1 + 13.8T + 73T^{2} \)
79 \( 1 - 12.0T + 79T^{2} \)
83 \( 1 - 9.00T + 83T^{2} \)
89 \( 1 - 7.22T + 89T^{2} \)
97 \( 1 - 2.29T + 97T^{2} \)
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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

−7.61979700856925993043626625161, −6.86467552317700987218721310721, −6.33234303810638309181672507773, −5.86793332476945793330792410577, −5.26316309898012284830976960649, −4.50556255577212382689002695394, −3.23877136692771085503896526409, −2.71155203316148887155414304824, −1.77376919841295182462719975837, −0.69038373908941894931825389365, 0.69038373908941894931825389365, 1.77376919841295182462719975837, 2.71155203316148887155414304824, 3.23877136692771085503896526409, 4.50556255577212382689002695394, 5.26316309898012284830976960649, 5.86793332476945793330792410577, 6.33234303810638309181672507773, 6.86467552317700987218721310721, 7.61979700856925993043626625161

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