# Properties

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

# Related objects

## Dirichlet series

 L(s)  = 1 + 1.41i·5-s + i·7-s + 1.41·11-s + 1.41i·17-s − 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s + 1.41i·41-s − 49-s + 2.00i·55-s + 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯
 L(s)  = 1 + 1.41i·5-s + i·7-s + 1.41·11-s + 1.41i·17-s − 2i·19-s − 1.41·23-s − 1.00·25-s − 1.41·35-s + 1.41i·41-s − 49-s + 2.00i·55-s + 1.41·71-s + 1.41i·77-s − 2.00·85-s − 1.41i·89-s + ⋯

## Functional equation

\begin{aligned} \Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}
\begin{aligned} \Lambda(s)=\mathstrut & 2016 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}

## Invariants

 $$d$$ = $$2$$ $$N$$ = $$2016$$    =    $$2^{5} \cdot 3^{2} \cdot 7$$ $$\varepsilon$$ = $-i$ motivic weight = $$0$$ character : $\chi_{2016} (1441, \cdot )$ primitive : yes self-dual : no analytic rank = 0 Selberg data = $(2,\ 2016,\ (\ :0),\ -i)$ $L(\frac{1}{2})$ $\approx$ $1.183122469$ $L(\frac12)$ $\approx$ $1.183122469$ $L(1)$ not available $L(1)$ not available

## Euler product

$L(s) = \prod_{p \text{ prime}} F_p(p^{-s})^{-1}$ where, for $p \notin \{2,\;3,\;7\}$, $$F_p$$ is a polynomial of degree 2. If $p \in \{2,\;3,\;7\}$, then $F_p$ is a polynomial of degree at most 1.
$p$$F_p$
bad2 $$1$$
3 $$1$$
7 $$1 - iT$$
good5 $$1 - 1.41iT - T^{2}$$
11 $$1 - 1.41T + T^{2}$$
13 $$1 - T^{2}$$
17 $$1 - 1.41iT - T^{2}$$
19 $$1 + 2iT - T^{2}$$
23 $$1 + 1.41T + T^{2}$$
29 $$1 + T^{2}$$
31 $$1 - T^{2}$$
37 $$1 + T^{2}$$
41 $$1 - 1.41iT - T^{2}$$
43 $$1 + T^{2}$$
47 $$1 - T^{2}$$
53 $$1 + T^{2}$$
59 $$1 - T^{2}$$
61 $$1 - T^{2}$$
67 $$1 + T^{2}$$
71 $$1 - 1.41T + T^{2}$$
73 $$1 - T^{2}$$
79 $$1 + T^{2}$$
83 $$1 - T^{2}$$
89 $$1 + 1.41iT - T^{2}$$
97 $$1 - 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

−9.525831406850213719209529330084, −8.796876090294785283832789724259, −8.024783850143527977247641519413, −6.95067350979800952610770344633, −6.41620939846779624137200615489, −5.87805167280411652657719349527, −4.57235103219738370142888263103, −3.62851429915875349752060224539, −2.75275375171096306086207964270, −1.83053504815339100853283845117, 0.911336349285537840221428926381, 1.85049143501637111046529227519, 3.69769869507050417664612806438, 4.10731287785860528203320251107, 5.04820317817149915537790032745, 5.91417226871951285500437089425, 6.81602805952662671527989733065, 7.72795850514979939633403928328, 8.338299410909366001810149279223, 9.243428147065903774952406887985