# Properties

 Degree 2 Conductor $2^{2} \cdot 37$ Sign $0.763 + 0.646i$ Motivic weight 0 Primitive yes Self-dual no Analytic rank 0

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## Dirichlet series

 L(s)  = 1 − i·3-s − 7-s + i·11-s + (−1 + i)17-s + (1 − i)19-s + i·21-s + (−1 + i)23-s + i·25-s − i·27-s + (−1 − i)29-s + 33-s − i·37-s − i·41-s + 47-s + (1 + i)51-s + ⋯
 L(s)  = 1 − i·3-s − 7-s + i·11-s + (−1 + i)17-s + (1 − i)19-s + i·21-s + (−1 + i)23-s + i·25-s − i·27-s + (−1 − i)29-s + 33-s − i·37-s − i·41-s + 47-s + (1 + i)51-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 148 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.763 + 0.646i)\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$148$$    =    $$2^{2} \cdot 37$$ $$\varepsilon$$ = $0.763 + 0.646i$ motivic weight = $$0$$ character : $\chi_{148} (105, \cdot )$ primitive : yes self-dual : no analytic rank = $$0$$ Selberg data = $$(2,\ 148,\ (\ :0),\ 0.763 + 0.646i)$$ $$L(\frac{1}{2})$$ $$\approx$$ $$0.6069956411$$ $$L(\frac12)$$ $$\approx$$ $$0.6069956411$$ $$L(1)$$ not available $$L(1)$$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$where, for $p \notin \{2,\;37\}$,$$F_p(T)$$ is a polynomial of degree 2. If $p \in \{2,\;37\}$, then $F_p(T)$ is a polynomial of degree at most 1.
$p$$F_p(T)$
bad2 $$1$$
37 $$1 + iT$$
good3 $$1 + iT - T^{2}$$
5 $$1 - iT^{2}$$
7 $$1 + T + T^{2}$$
11 $$1 - iT - T^{2}$$
13 $$1 - iT^{2}$$
17 $$1 + (1 - i)T - iT^{2}$$
19 $$1 + (-1 + i)T - iT^{2}$$
23 $$1 + (1 - i)T - iT^{2}$$
29 $$1 + (1 + i)T + iT^{2}$$
31 $$1 + iT^{2}$$
41 $$1 + iT - T^{2}$$
43 $$1 - iT^{2}$$
47 $$1 - T + T^{2}$$
53 $$1 - T + T^{2}$$
59 $$1 - iT^{2}$$
61 $$1 + iT^{2}$$
67 $$1 - T^{2}$$
71 $$1 - T + T^{2}$$
73 $$1 + iT - T^{2}$$
79 $$1 + (1 - i)T - iT^{2}$$
83 $$1 + T + T^{2}$$
89 $$1 + (1 + i)T + iT^{2}$$
97 $$1 - iT^{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

−13.10816457337357686041792780832, −12.43625305431146567598223854948, −11.37649815088232660667564705338, −9.990264392653704431458007765550, −9.141281680914141676207393897133, −7.56795461850848674019017603429, −6.94968605451004120504551546434, −5.77021853794458276231113710756, −3.95290976747288843243538632860, −2.08628502080588857118603629212, 3.09928043927083506759952653755, 4.27714347876596393041059830264, 5.69019913107289534205610998386, 6.89050957330215446206173448352, 8.444784212317855955232376186194, 9.522405592957802838890433163166, 10.19320403310866793307351058503, 11.23585659445774750887200017992, 12.38211825179779823233357558045, 13.49148832149289409247800683737