# Properties

 Degree 2 Conductor $2^{2} \cdot 7 \cdot 11 \cdot 13$ Sign $-0.361 + 0.932i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

# Related objects

## Dirichlet series

 L(s)  = 1 − 1.07·3-s − 3.26i·5-s + i·7-s − 1.84·9-s − i·11-s + (1.30 − 3.36i)13-s + 3.50i·15-s + 3.05·17-s + 3.15i·19-s − 1.07i·21-s + 5.49·23-s − 5.63·25-s + 5.20·27-s + 4.12·29-s − 6.96i·31-s + ⋯
 L(s)  = 1 − 0.621·3-s − 1.45i·5-s + 0.377i·7-s − 0.614·9-s − 0.301i·11-s + (0.361 − 0.932i)13-s + 0.905i·15-s + 0.740·17-s + 0.724i·19-s − 0.234i·21-s + 1.14·23-s − 1.12·25-s + 1.00·27-s + 0.766·29-s − 1.25i·31-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 4004 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.361 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$4004$$    =    $$2^{2} \cdot 7 \cdot 11 \cdot 13$$ $$\varepsilon$$ = $-0.361 + 0.932i$ motivic weight = $$1$$ character : $\chi_{4004} (2157, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 4004,\ (\ :1/2),\ -0.361 + 0.932i)$ $L(1)$ $\approx$ $1.320763303$ $L(\frac12)$ $\approx$ $1.320763303$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;7,\;11,\;13\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;7,\;11,\;13\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
7 $$1 - iT$$
11 $$1 + iT$$
13 $$1 + (-1.30 + 3.36i)T$$
good3 $$1 + 1.07T + 3T^{2}$$
5 $$1 + 3.26iT - 5T^{2}$$
17 $$1 - 3.05T + 17T^{2}$$
19 $$1 - 3.15iT - 19T^{2}$$
23 $$1 - 5.49T + 23T^{2}$$
29 $$1 - 4.12T + 29T^{2}$$
31 $$1 + 6.96iT - 31T^{2}$$
37 $$1 - 10.1iT - 37T^{2}$$
41 $$1 - 0.829iT - 41T^{2}$$
43 $$1 - 10.0T + 43T^{2}$$
47 $$1 + 9.75iT - 47T^{2}$$
53 $$1 - 7.58T + 53T^{2}$$
59 $$1 + 15.0iT - 59T^{2}$$
61 $$1 + 14.7T + 61T^{2}$$
67 $$1 - 9.94iT - 67T^{2}$$
71 $$1 + 14.2iT - 71T^{2}$$
73 $$1 - 7.06iT - 73T^{2}$$
79 $$1 - 11.7T + 79T^{2}$$
83 $$1 + 0.295iT - 83T^{2}$$
89 $$1 - 8.17iT - 89T^{2}$$
97 $$1 + 9.48iT - 97T^{2}$$
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\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

−8.265928503372400476162704077973, −7.78445485907508464975755910454, −6.49051672176223964527780720876, −5.78940122751823949095969023772, −5.30850304632614661560052681985, −4.72737509458732552191544390605, −3.61696793270652364710984626711, −2.70137019267261210021494513803, −1.26665997357694286251826888908, −0.52766596524892304750769902346, 1.06497103971239074793356454567, 2.51777834096143693378849808888, 3.10492560513614944198086989533, 4.09050637990791495997131913298, 4.98464463077447387485997512867, 5.89843580956891437007583129362, 6.49957544571526193354266123111, 7.14054689949637653471933057693, 7.61002762907543996522168723157, 8.842709618172817386223949210915