Properties

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

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  − 3-s − 3.06·5-s + 3.59·7-s + 9-s + 3.57·11-s + 1.11·13-s + 3.06·15-s − 5.35·17-s − 1.67·19-s − 3.59·21-s + 23-s + 4.38·25-s − 27-s − 29-s + 5.98·31-s − 3.57·33-s − 11.0·35-s − 4.76·37-s − 1.11·39-s + 3.76·41-s − 11.7·43-s − 3.06·45-s − 6.41·47-s + 5.90·49-s + 5.35·51-s − 0.261·53-s − 10.9·55-s + ⋯
L(s)  = 1  − 0.577·3-s − 1.37·5-s + 1.35·7-s + 0.333·9-s + 1.07·11-s + 0.308·13-s + 0.791·15-s − 1.29·17-s − 0.383·19-s − 0.784·21-s + 0.208·23-s + 0.877·25-s − 0.192·27-s − 0.185·29-s + 1.07·31-s − 0.621·33-s − 1.86·35-s − 0.783·37-s − 0.178·39-s + 0.588·41-s − 1.79·43-s − 0.456·45-s − 0.935·47-s + 0.844·49-s + 0.749·51-s − 0.0359·53-s − 1.47·55-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: \(1\)
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 \( 1 + 3.06T + 5T^{2} \)
7 \( 1 - 3.59T + 7T^{2} \)
11 \( 1 - 3.57T + 11T^{2} \)
13 \( 1 - 1.11T + 13T^{2} \)
17 \( 1 + 5.35T + 17T^{2} \)
19 \( 1 + 1.67T + 19T^{2} \)
31 \( 1 - 5.98T + 31T^{2} \)
37 \( 1 + 4.76T + 37T^{2} \)
41 \( 1 - 3.76T + 41T^{2} \)
43 \( 1 + 11.7T + 43T^{2} \)
47 \( 1 + 6.41T + 47T^{2} \)
53 \( 1 + 0.261T + 53T^{2} \)
59 \( 1 + 7.33T + 59T^{2} \)
61 \( 1 + 3.81T + 61T^{2} \)
67 \( 1 + 0.212T + 67T^{2} \)
71 \( 1 + 15.4T + 71T^{2} \)
73 \( 1 - 15.2T + 73T^{2} \)
79 \( 1 - 14.1T + 79T^{2} \)
83 \( 1 - 4.48T + 83T^{2} \)
89 \( 1 - 7.32T + 89T^{2} \)
97 \( 1 - 5.66T + 97T^{2} \)
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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

−7.59004784354740028972458774381, −6.68254208416003621508827920837, −6.37111003789966258811047327800, −5.12632240171334015470485569935, −4.61503029013099577786712202141, −4.10958888315884158621668616921, −3.34975702466916400659444519824, −2.00856249862205444391266130418, −1.18222793431395782091392522428, 0, 1.18222793431395782091392522428, 2.00856249862205444391266130418, 3.34975702466916400659444519824, 4.10958888315884158621668616921, 4.61503029013099577786712202141, 5.12632240171334015470485569935, 6.37111003789966258811047327800, 6.68254208416003621508827920837, 7.59004784354740028972458774381

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