# Properties

 Degree 2 Conductor $2 \cdot 3^{2} \cdot 5^{2} \cdot 7$ Sign $-0.447 - 0.894i$ Motivic weight 1 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 + i·2-s − 4-s + i·7-s − i·8-s + 4·11-s − 2i·13-s − 14-s + 16-s + 2i·17-s − 4·19-s + 4i·22-s + 8i·23-s + 2·26-s − i·28-s − 2·29-s + ⋯
 L(s)  = 1 + 0.707i·2-s − 0.5·4-s + 0.377i·7-s − 0.353i·8-s + 1.20·11-s − 0.554i·13-s − 0.267·14-s + 0.250·16-s + 0.485i·17-s − 0.917·19-s + 0.852i·22-s + 1.66i·23-s + 0.392·26-s − 0.188i·28-s − 0.371·29-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(2-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 3150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$3150$$    =    $$2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ $$\varepsilon$$ = $-0.447 - 0.894i$ motivic weight = $$1$$ character : $\chi_{3150} (2899, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 3150,\ (\ :1/2),\ -0.447 - 0.894i)$ $L(1)$ $\approx$ $1.589368584$ $L(\frac12)$ $\approx$ $1.589368584$ $L(\frac{3}{2})$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;3,\;5,\;7\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;5,\;7\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1 - iT$$
3 $$1$$
5 $$1$$
7 $$1 - iT$$
good11 $$1 - 4T + 11T^{2}$$
13 $$1 + 2iT - 13T^{2}$$
17 $$1 - 2iT - 17T^{2}$$
19 $$1 + 4T + 19T^{2}$$
23 $$1 - 8iT - 23T^{2}$$
29 $$1 + 2T + 29T^{2}$$
31 $$1 + 31T^{2}$$
37 $$1 + 6iT - 37T^{2}$$
41 $$1 - 6T + 41T^{2}$$
43 $$1 + 4iT - 43T^{2}$$
47 $$1 - 47T^{2}$$
53 $$1 - 10iT - 53T^{2}$$
59 $$1 - 12T + 59T^{2}$$
61 $$1 - 14T + 61T^{2}$$
67 $$1 - 12iT - 67T^{2}$$
71 $$1 - 8T + 71T^{2}$$
73 $$1 - 10iT - 73T^{2}$$
79 $$1 + 16T + 79T^{2}$$
83 $$1 - 12iT - 83T^{2}$$
89 $$1 - 10T + 89T^{2}$$
97 $$1 + 2iT - 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

−8.826720435876726295317263037843, −8.212943581389159326109155280598, −7.33013367924308343034170547799, −6.73308660960340395096537949757, −5.78842418283782846665269507767, −5.45177830962152926500253079278, −4.13904590080917166872358906659, −3.71753193214688542609658064791, −2.35889104555269450690370290147, −1.13239738017760799271677146140, 0.56464863805919409306505491886, 1.73257711146949815551442522475, 2.66116301738036995538247288159, 3.79992080817952944918219692036, 4.31857800866127595740348614604, 5.13370250830456321850211520648, 6.45051981026895309423706895960, 6.67989243994877390682436296510, 7.85087508873386847634823185653, 8.647070953951551054909897349010