# Properties

 Degree 2 Conductor $2^{2} \cdot 3 \cdot 23 \cdot 29$ Sign $-1$ Motivic weight 1 Primitive yes Self-dual yes Analytic rank 1

# Related objects

## Dirichlet series

 L(s)  = 1 + 3-s − 0.270·5-s + 0.409·7-s + 9-s − 3.18·11-s + 0.382·13-s − 0.270·15-s − 4.23·17-s + 5.44·19-s + 0.409·21-s + 23-s − 4.92·25-s + 27-s + 29-s + 2.08·31-s − 3.18·33-s − 0.110·35-s − 5.55·37-s + 0.382·39-s + 7.07·41-s − 5.46·43-s − 0.270·45-s − 11.6·47-s − 6.83·49-s − 4.23·51-s + 13.1·53-s + 0.861·55-s + ⋯
 L(s)  = 1 + 0.577·3-s − 0.121·5-s + 0.154·7-s + 0.333·9-s − 0.959·11-s + 0.106·13-s − 0.0699·15-s − 1.02·17-s + 1.24·19-s + 0.0892·21-s + 0.208·23-s − 0.985·25-s + 0.192·27-s + 0.185·29-s + 0.373·31-s − 0.553·33-s − 0.0187·35-s − 0.913·37-s + 0.0612·39-s + 1.10·41-s − 0.833·43-s − 0.0403·45-s − 1.70·47-s − 0.976·49-s − 0.592·51-s + 1.80·53-s + 0.116·55-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

 $$d$$ = $$2$$ $$N$$ = $$8004$$    =    $$2^{2} \cdot 3 \cdot 23 \cdot 29$$ $$\varepsilon$$ = $-1$ motivic weight = $$1$$ character : $\chi_{8004} (1, \cdot )$ primitive : yes self-dual : yes analytic rank = 1 Selberg data = $(2,\ 8004,\ (\ :1/2),\ -1)$ $L(1)$ $=$ $0$ $L(\frac12)$ $=$ $0$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;23,\;29\}$, $F_p(T) = 1 - a_p T + p T^2 .$If $p \in \{2,\;3,\;23,\;29\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1 - T$$
23 $$1 - T$$
29 $$1 - T$$
good5 $$1 + 0.270T + 5T^{2}$$
7 $$1 - 0.409T + 7T^{2}$$
11 $$1 + 3.18T + 11T^{2}$$
13 $$1 - 0.382T + 13T^{2}$$
17 $$1 + 4.23T + 17T^{2}$$
19 $$1 - 5.44T + 19T^{2}$$
31 $$1 - 2.08T + 31T^{2}$$
37 $$1 + 5.55T + 37T^{2}$$
41 $$1 - 7.07T + 41T^{2}$$
43 $$1 + 5.46T + 43T^{2}$$
47 $$1 + 11.6T + 47T^{2}$$
53 $$1 - 13.1T + 53T^{2}$$
59 $$1 - 4.76T + 59T^{2}$$
61 $$1 + 2.32T + 61T^{2}$$
67 $$1 - 0.562T + 67T^{2}$$
71 $$1 + 7.04T + 71T^{2}$$
73 $$1 + 10.9T + 73T^{2}$$
79 $$1 + 16.6T + 79T^{2}$$
83 $$1 - 9.44T + 83T^{2}$$
89 $$1 - 8.74T + 89T^{2}$$
97 $$1 + 2.87T + 97T^{2}$$
show less
\begin{aligned} L(s) = \prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1} \end{aligned}

## Imaginary part of the first few zeros on the critical line

−7.54837484346023481190729641668, −6.98170240556482792206651671750, −6.12105604139399049691432521071, −5.28686806129007643823938778017, −4.69192721276418008809960885739, −3.82518081848001872219409461030, −3.04788441250730363541327967364, −2.33414834708812021740703344094, −1.38608665368054335285387118348, 0, 1.38608665368054335285387118348, 2.33414834708812021740703344094, 3.04788441250730363541327967364, 3.82518081848001872219409461030, 4.69192721276418008809960885739, 5.28686806129007643823938778017, 6.12105604139399049691432521071, 6.98170240556482792206651671750, 7.54837484346023481190729641668