# Properties

 Degree 2 Conductor $2^{2} \cdot 89$ Sign $1$ Motivic weight 0 Primitive yes Self-dual yes Analytic rank 0

# Origins

## Dirichlet series

 L(s)  = 1 + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯
 L(s)  = 1 + 2-s − 3-s + 4-s − 5-s − 6-s + 2·7-s + 8-s − 10-s − 12-s + 2·14-s + 15-s + 16-s − 17-s − 19-s − 20-s − 2·21-s − 23-s − 24-s + 27-s + 2·28-s + 30-s − 31-s + 32-s − 34-s − 2·35-s − 38-s − 40-s + ⋯

## Functional equation

\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}
\begin{aligned}\Lambda(s)=\mathstrut & 356 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$356$$    =    $$2^{2} \cdot 89$$ $$\varepsilon$$ = $1$ motivic weight = $$0$$ character : $\chi_{356} (355, \cdot )$ primitive : yes self-dual : yes analytic rank = 0 Selberg data = $(2,\ 356,\ (\ :0),\ 1)$ $L(\frac{1}{2})$ $\approx$ $1.058354802$ $L(\frac12)$ $\approx$ $1.058354802$ $L(1)$ not available $L(1)$ not available

## Euler product

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

−11.67664387409176254961069404999, −11.12641832666094511225923344933, −10.59630604998005499570730013389, −8.486875419216600821463090171005, −7.81465077644172741318499151707, −6.72180728980827205506652916631, −5.57538027644770783049218703171, −4.72657548264463000766072728667, −4.01600802562592568972815723771, −2.00491779343604680931144269321, 2.00491779343604680931144269321, 4.01600802562592568972815723771, 4.72657548264463000766072728667, 5.57538027644770783049218703171, 6.72180728980827205506652916631, 7.81465077644172741318499151707, 8.486875419216600821463090171005, 10.59630604998005499570730013389, 11.12641832666094511225923344933, 11.67664387409176254961069404999